WO2006018034A1

WO2006018034A1 - Filter apparatus and method for frequency domain filtering

Info

Publication number: WO2006018034A1
Application number: PCT/EP2004/009372
Authority: WO
Inventors: Günther Auer
Original assignee: Ntt Docomo, Inc.
Priority date: 2004-08-20
Filing date: 2004-08-20
Publication date: 2006-02-23

Abstract

An apparatus for frequency domain filtering comprises a transformer (101) for transforming a time domain signal into a frequency domain signal, a low-pass filter (107) for filtering the frequency domain signal, the low-pass filter (107) having real-valued coefficients. The transformer (101) is configured for pre-processing the time domain signal or for processing the frequency domain signal, in order to introduce a phase shift to the transformed signal so as to shift the spectrum of the transformed signal towards the pass-band of the low-pass filter (107). Thus, the present invention provides for low-complexity filtering in frequency domain.

Description

Filter apparatus and method for frequency domain filtering

The present invention is in the field of digital signal processing and, in particular, in the field of digital fil- tering in frequency domain.

Multi-carrier modulation, in particular orthogonal fre¬ quency division multiplexing (OFDM) has been successfully applied to transmitting information in a plurality of digi- tal communication systems. In order to obtain an OFDM sig¬ nal to be transmitted, a plurality of data values compris¬ ing the information to be transmitted is mapped onto a num¬ ber of complex symbols in accordance with a chosen mapping scheme, e.g. QAM (QAM = Quadrature Amplitude Modulation) . The number of complex values are further processed using e.g. an Inverse Fourier Transform (IFT) . Therefore, the number of complex values representing the plurality of data values is considered as being a frequency domain signal since each complex value is associated with a sub-carrier appointed to a certain frequency point.

Fig. 13 shows a block diagram of an OFDM receiver utilizing a Fast Fourier Transform (FFT) for time-frequency trans¬ forming. By inserting a cyclic prefix into the guard inter- val (GI) longer than the maximum delay of the channel through which the resulting OFDM signal is to be transmit¬ ted, intersymbol interference (ISI) can be eliminated com¬ pletely and the orthogonality of the received signal is preserved.

As is shown in Fig. 13, the received, so called time domain signal is transformed into a transformed signal in the so- called frequency domain by the means of e.g. the Fast Fou¬ rier Transform. The transformed signal represents a re- ceived version of the number of complex values obtained from mapping data values onto signal space constellation points. In other words, the transformed signal represents a one-sided spectrum of the received time domain OFDM signal. In addition, the transformed signal being a received ver¬ sion of the number of complex values comprises at least partly the transmitted information. Therefore, the trans- formed signal in the so-called frequency domain is a signal having a certain spectrum. If, for example, a Fourier Transform is applied for transforming the transformed sig¬ nal, then a spectrum of the transformed signal obtained from the Fourier Transform is, due to Fourier Transform rules, closely related to the received time domain signal.

In frequency domain, an influence of the communication channel can be described by a channel transfer function, which, generally, is complex. In Fig. 14, by the way of ex- ample only, a phase of a channel transfer function and a magnitude of the channel transfer function are shown. Fig. 15 shows a corresponding spectrum of the channel transfer function of Fig. 14, wherein the corresponding spectrum is the time domain channel impulse response being only non-zero within the range left [0, τ_maχ] , where τ_max de¬ notes the maximum delay of the channel. From this point of view, the spectrum of the channel transfer function is one sided and has a band-pass characteristic. Therefore, proc¬ essing signals in the so-called "frequency domain" is often associated with complex signal processing structures, like for example complex filters, in order to e.g. equalize the received OFDM signal, corresponding to the transformed sig¬ nal mentioned above, in frequency domain. However, filter¬ ing complex valued signals using complex filters having complex valued coefficients is associated with a high com¬ putational complexity since complex convolution operations need to be performed.

In order to equalize the transformed signal, the channel transfer function is to be estimated. Usually, known pilot symbols are used in the transmitter for modulating sub- carriers, wherein, in a receiver, the corresponding re¬ ceived sub-carriers are de-modulated using the (known) pi- lot symbols in order to obtain sub-carrier values compris¬ ing information on the channel transfer function at fre¬ quency points being associated with sub-carriers being modulated by pilot symbols. Although the sub-carrier values already comprise an estimate of the channel transfer func¬ tion for the modulated sub-carriers, a further estimate of the channel transfer function with a better accuracy can be obtained when e.g. filtering the channel transfer estimate using e.g. Wiener filters. In P. Hδher, "TCM on Frequency Selective Land-Mobile Radio Channels," in Proc. 5th Tirre- nia Int. Workshop on Digital Communication, Tirrenia, It¬ aly, pages 317-328, September 1991, Wiener filtering in frequency domain is described where the coefficients of the Wiener filter are determined on the basis of determining an correlation function of a channel process.

Fig. 16 shows a block diagram of an OFDM receiver wherein, after FFT, the sub-carriers being modulated by pilot sym¬ bols are de-multiplexed (DMUX pilot) and provided to a channel estimation unit being configured for estimating the channel transfer function on a basis of the de-multiplexed sub-carriers. The estimator of the channel transfer func¬ tion is then provided to a means for extracting information (DET) , the means for extracting information being config- ured e.g. for equalizing the frequency domain signal using the estimate of the channel transfer function. However, since only certain sub-carriers are modulated with pilot symbols in the transmitter, the estimate of the channel transfer function does not comprise any information on channel transfer coefficients at frequency points associ¬ ated with sub-carriers, which are not used for pilot symbol transmission. In order to obtain an estimate of the channel transfer function covering all frequency points used for transmission, the estimate of the channel transfer function can be interpolated using e.g. interpolation filters. How¬ ever, this operation is associated with an increased com¬ plexity since the interpolation filter are complex in order to interpolate between complex values of a signal having a band-pass characteristic.

It is the object of the present invention to provide a con- cept for frequency domain filtering with reduced complex¬ ity.

This object is achieved by the filter apparatus according to claim 1 or by a multi-carrier receiver according to claim 21 or by a method for frequency domain filtering ac¬ cording to claim 22 or by a method for receiving multi- carrier signals according to claim 23 or by a computer pro¬ gram according to claim 24.

The present invention is based on the finding that complex¬ ity reduced frequency domain filtering can be achieved when using filters having real valued coefficients only, pro¬ vided that a spectrum of a frequency received signal is shifted towards a pass-band of the filter.

In particular, the present invention exploits the fact that a spectrum of a real valued filter response is two-sided and symmetric. In order to shift a spectrum of the trans¬ formed signal towards the pass-band of the filter for low- complexity filtering, the transformed signal resulting from time-frequency transforming a time domain signal, either the time domain signal is pre-processed in time domain, or the transformed signal is post-processed in frequency do¬ main, or the time domain signal is pre-processed in time domain and the transformed signal is post-processed in fre¬ quency domain. Preferably, the time domain signal is de¬ layed in time domain, for example cyclically delayed, so that a phase shift is introduced to the transformed signal resulting from time-frequency transforming the time domain signal after pre-processing. Furthermore, a phase of the transformed signal may be post-processed in order to di¬ rectly introduce a phase shift to the transformed signal for influencing its spectrum. It is an advantage of the present invention that, in fre¬ quency domain, real valued filter can be used for filter¬ ing, which significantly reduces e.g. a receiver's complex- ity.

It is a further advantage of the present invention that a width of a pass-band of the frequency domain filter may be chosen, e.g. by approximating a rectangular shape, such that a width of the pass-band optimally matches a spectral range occupied by the transformed signal to be filtered or even coincides with the spectral range so that an accurate frequency domain filtering can be performed.

Further embodiments of the present invention are described with regard to the following figures, in which:

Fig. 1 shows a block diagram of a filter apparatus for frequency domain filtering in accordance with an embodiment of the present invention;

Fig. 2a shows the inventive post-processing approach;

Fig. 2b shows the inventive pre-processing approach;

Fig. 3a shows a phase of a channel transfer function;

Fig. 3b shows a magnitude of a channel transfer function;

Fig. 4 shows a corresponding spectrum of the channel transfer function of Figs. 3a and 3b;

Fig. 5 shows a block diagram of an apparatus for reduc¬ ing a phase drift in accordance with a further embodiment of the present invention; Fig. 6 shows a block diagram of an apparatus for reduc¬ ing a phase drift in accordance with a further embodiment o'f the present invention;

Fig. 7 shows OFDM system parameters;

Fig. 8 shows a power delay profile of a channel;

Fig. 9 demonstrates a performance of the inventive ap- proach;

Fig. 10 shows an effective channel impulse response re¬ sulting when cyclically shifting a received sig¬ nal in time domain;

Fig. 11 demonstrates the performance of the inventive ap¬ proach;

Fig. 12 shows a block diagram of the inventive apparatus for reducing a phase shift in accordance with a further embodiment of the present invention;

Fig. 13 shows a block diagram of an OFDM receiver;

Fig. 14 shows a phase and a magnitude of a channel trans¬ form functions;

Fig. 15 shows a corresponding time domain channel impulse response; and

Fig. 16 shows a channel estimation approach.

Fig. 1 shows a block diagram of a filter apparatus for fre¬ quency domain filtering in accordance with an embodiment of the present invention.

The filter apparatus comprises a transformer 101, the transformer 101 having an input 103 and an output 105, the output 105 of the transformer 101 being coupled to a low- pass filter 107, the low-pass filter 107 having an output 109.

The transformer 101 is configured for time-frequency trans¬ forming a time domain signal being provided to the trans¬ former 101 via the input 103 into a transformed signal into frequency domain, the transformed signal being provided via the output 105 of the transformer 101 to the low-pass fil- ter 107.

In accordance with the present invention, the low-pass fil¬ ter 107 comprises only real-valued filter coefficients rep¬ resenting the real part of a filter response, wherein fil- ter coefficients representing an imaginary part of the fil¬ ter response are set to zero. Preferably, the filter has a filter response, which is only real valued.

In accordance with the present invention, the transformer is configured for pre-processing the signal before time- frequency transforming, i.e. for pre-processing the time domain signal, or for post-processing the transformed sig¬ nal before filtering, the transformed signal 'resulting from time-frequency transforming the time domain signal in order to introduce a phase shift to the transformed signal for shifting a spectrum of the transformed signal towards a pass-band of the low-pass filter.

In order to demonstrate the inventive approach, Fig. 1 fur- ther shows a magnitude M of a transformed signal 111 pro¬ vided by the transformer 105 after pre- or post-processing, In accordance with the present invention, the transformed signal being provided by the transformer 101 has a spectrum 113, which spectrum is two-sided and symmetric with respect to an origin of an axis f' . In accordance with the present invention, a spectrum 115 of a transformed signal prior to pre- or post-processing is shifted in the shift direction as depicted by the arrow towards a pass-band 117 of the low-pass filter 107, wherein the pass-band 117 is symmetric and two-sided with respect to the origin of the axis f , wherein M' denotes a magnitude in the f' region.

As is shown in Fig. 1, the transformer 101 is configured for pre-processing the time domain signal and/or for post¬ processing the transformed signal after time-frequency transform in order to manipulate the spectrum of the re¬ sulting transformed signal in the frequency domain f .

It is to be noted that, in the so-called frequency domain f, the transformed signal may be considered as a set of values, wherein each value of the transformed signal is as¬ sociated with a frequency point. Nevertheless, the set of values represents a signal which can be filtered using a convolution operation. Since the filter coefficients are real valued only, a complex convolution operation can be avoided.

In accordance with a further embodiment of the present in¬ vention, the transformer 101 is configured for pre¬ processing the time domain signal or for post-processing the transformed signal in order to shift the spectrum of the transformed signal by a spectrum distance of half the width of the pass-band of the low-pass filter 107, the pass-band being symmetric with respect to an origin in the f region.

Alternatively, the transformer 101 may be configured for pre-processing the time domain signal or for post¬ processing the transformed signal in order to shift the spectrum of the unprocessed transformed signal by a spec¬ trum distance of half the band width of the transformed signal. For example, if the transformed signal prior to processing has a one-sided spectrum having a band-pass characteristic, then the transformer is configured for shifting the spectrum of the transformed signal such that a resulting spectrum is symmetric with respect to the origin, so that the spectrum of the resulting transformed signal coincides with the pass-band of the low-pass filter 107 or, at least, overlaps with the pass-band.

For example, the time domain signal is a received version of a transmit signal, the transmit signal being transmitted through a communication channel from a transmitting point to a receiving point, the receiving point comprising the inventive apparatus for frequency domain filtering. Pref- erably, the transmit signal is a time domain transmit sig¬ nal resulting from frequency-time transforming a multi- carrier signal in frequency domain for multi-carrier trans¬ mission, for example, for multi-carrier wireless transmis¬ sion.

In accordance with a further aspect of the present inven¬ tion, a width of the pass-band of the low-pass filter 107 may be determined by a maximum channel delay associated with the communication channel through which the transmit signal is to be transmitted or by a length of a power delay profile associated with that channel.

In accordance with a further aspect of the present inven¬ tion, the width of the pass-band of the low-pass filter 107 may be determined by a channel delay of a certain communi¬ cation channel, the certain communication channel having a maximum channel delay among a plurality of communication channels to be considered when receiving the transmit sig¬ nal. For example, each communication channel may be repre- sented by a certain type of communication channel being as¬ sociated with a certain type of environment in case of wireless transmission or associated with a certain cable length in a case of a wired transmission. For example, in the case of a wireless transmission, a plurality of commu- nication channels is possible for the different kinds of environment like e.g. urban areas or hilly terrains. If, for example, a wireless receiver comprising the inventive filter apparatus is mobile, then the a plurality of commu- nication channels associated with a plurality of receiving scenarios is possible. In order to take this situation into account, the width of the pass-band may be determined by a maximum expectable channel delay or, in other words, by a longest power delay profile specifying an attenuation of signal components over time.

For example, the width of the pass-band of the low-pass filter 107 is equal to or smaller than the maximum channel delay plus an additional delay introduced by a slope of a filter response in order to take a finite slope duration of the filter response into account.

In accordance with the present invention, the pass-band of the filter 107 may be uniformly shaped and have, for exam¬ ple, an approximately rectangular shape. However, the pass- band of the low-pass filter 107 may have any shape provided that the pass-band is symmetric with respect to an origin of a spectral region (e.g. to an origin of the f axis de- picted in Fig. 1) .

In order to shift the spectrum of the transformed signal towards the pass-band of the low-pass filter 107, the transformer is configured for pre-processing the time do- main signal in time domain and/or for post-processing the transformed signal in frequency domain so that a phase shift is introduced into the transformed signal.

In order to introduce the phase shift to the transformed signal, the transformer may be configured for delaying the time domain signal in time domain, the phase shift in fre¬ quency domain introducing a shift of the spectrum of the transformed signal, e.g. along the f axis.

For example, the transformer 101 may be configured for cy¬ clically delaying or for cyclically shifting the time do¬ main signal. In order to cyclically shifting the time do¬ main signal, the transformer 101 may comprise a cyclic shift element for pre-processing the time domain signal, the cyclic shift element being configured for cyclically shifting the time domain signal by a number of values in order to introduce a phase shift to the transformed signal in frequency domain. For example, the number of values may be dependent on a width of the pass-band of the low-pass filter 107 in order to shift the spectrum of the trans¬ formed signal in dependence on the pass-band width of the low-pass filter.

Cyclically shifting the time domain signal means that val¬ ues of the time domain signal are shifted such that, in the case of a right shift, a last value of the time domain sig¬ nal is placed before a first value of the time domain sig- nal or, in the case of a left shift, that the first value of the time domain signal is attached at the end of the time domain signal and that the second value of the time domain signal becomes a first value of the time domain sig¬ nal.

In accordance with a further aspect of the present inven¬ tion, the number of values may be equal to or smaller than half the width of the pass-band in order to, for example, shift the spectrum of the transformed signal by the previ- ously mentioned spectrum distance of half the width of the pass-band of the low-pass filter 107.

In accordance with a further aspect of the present inven¬ tion, the number of values the time domain signal is to be shifted by is equal to half the difference between the width of the pass-band and an additional delay introduced by a slope of a filter response, when the slope of the fil¬ ter response has e.g. a finite duration.

In order to perform the cyclical shift, the transformer 101 may comprise a shift register, the shift register having an input and an output, wherein the output is coupled back to the input for cyclically shifting the time domain signal. For example, the shift register may be controllable by the transformer in dependence on a currently adjusted pass-band width of the low-pass filter 107.

In order to transform the time domain signal into frequency domain, the transformer 101 may comprise a Fourier trans¬ former, the Fourier transformer being configured for per¬ forming e.g. a Fast Fourier Transform (FFT) . Preferably, the output of the cyclic shift element is coupled to the Fourier transformer, wherein the Fourier transformer per¬ forms e.g. a serial to parallel conversion and the Fourier Transform, wherein, after having performed the Fourier Transform, a set of values representing the transformed signal is provided to the filter 107 for filtering.

In accordance with a further aspect of the present inven¬ tion, the transformer 101 may be configured for post¬ processing the transformed signal in order to directly in¬ troduce the phase shift to the transformed signal for shifting its spectrum. For example, the transformer 101 may be configured for time-frequency transforming the time do¬ main signal without pre-processing the time domain signal, so that the phase shift is introduced only in frequency do¬ main.

In accordance with a further aspect of the present inven¬ tion, the transformer may be configured for performing both: pre-processing in time domain and post-processing in frequency domain for an accurate spectrum shift. For exam- pie, the transformer 101 may be configured for cyclically shifting the time domain signal in order to introduce a coarse phase shift, an for post-processing the resulting transformed signal in frequency domain for introducing a fine phase shift so that the spectrum of the resulting transformed signal may accurately be positioned within the pass-band of the low-pass filter 107. Generally, the transformer 101 may be configured for chang¬ ing a phase of the transformed signal by e.g. multiplying each value of the transformed signal by a complex factor introducing the phase shift.

In order to post-process the transformed signal, the trans¬ former 101 may comprise a phase compensator being config¬ ured for phase shifting the transformed signal. For exam¬ ple, the phase shift may be dependent on half the width of the pass-band of the low-pass filter 107 in order to "fit" the spectrum of the transformed signal into the pass-band.

Preferably, an output of the phase compensator is connected to an input of the filter 107. Moreover, if a Fourier transformer is used for time-frequency conversion, then the outputs of the Fourier transformer may be connected to an input or to a plurality of inputs of the phase compensator for post-processing.

In accordance with a further aspect of the present inven¬ tion, the inventive apparatus for frequency domain filter¬ ing may be used in a receiver, preferably in a multi- carrier receiver, in order to process the time domain sig¬ nal, which is for example, a received version of a transmit signal being transmitted through a communication channel.

In accordance with a further aspect of the present inven¬ tion, the low-pass filter 107 may be configured for channel estimation, i.e. for providing an estimate of the channel transfer function from the transformed signal.

For example, the transformed signal may comprise a set of received versions of sub-carrier values resulting from modulating sub-carrier values by pilot symbols for channel estimation, wherein the pilot symbols are known in the transmitter and in the receiver. For example, the trans¬ former 101 may be configured for de-modulating the received versions of sub-carrier values using the pilot symbols in order to provide a set of values to the low-pass filter for channel estimation, the set of values representing the transformed signal. In order to de-modulate the received versions of sub-carrier values, the transformer 101 may be configured for dividing each received version of sub- carrier values by an associated pilot symbol or by multi¬ plying each received version of sub-carrier values by a complex conjugate version of an associated pilot symbol.

Moreover, after de-modulating, the received versions of sub-carrier values may be considered as being a coarse (or, in other words, a first) estimate of the channel transfer function, so that the transformed signal, i.e. the set of values, comprises a first estimate of the channel transfer function. The low-pass filter 107 may be configured for further estimating the channel, i.e. the channel transfer function, on a basis of the set of values in order to ob¬ tain a better estimate of the channel transfer function.

In accordance with a further aspect of the present inven¬ tion, the low-pass filter 107 is configured for de¬ modulating the received versions of sub-carrier values us¬ ing the pilot symbols in order to obtain the set of values as a coarse estimate of the channel transfer function, and for further estimating the channel on the basis of the set of values in order to provide a better, i.e. a more accu¬ rate estimate of the channel.

For example, the low-pass filter 107 comprises a Wiener filter for performing the channel estimation. Preferably, the low-pass filter comprises real valued coefficients be¬ ing obtained from solving a Wiener-Hopf equation. This is¬ sue will be addressed later in detail.

In accordance with a further aspect of the present inven¬ tion, the low-pass filter 107 my comprise an interpolation filter. For example, the interpolation filter is configured for performing an interpolation between two subsequent val- ues of an estimate of the channel transfer function in or¬ der to obtain interpolated values of the channel transfer function, when e.g. only certain sub-carriers were modu¬ lated by pilot symbols, so that sub-carriers between modu- lated sub-carriers cannot be exploited for estimating coef¬ ficients of the channel transfer function at the corre¬ sponding frequency points.

Accordingly, the low-pass filter 107 may be configured for interpolating between values in the set of values in order to obtain an estimate of the channel transfer function for all sub-carriers.

In accordance with a further aspect of the present inven- tion, the apparatus for frequency domain filtering may fur¬ ther comprise means for determining filter coefficients. For example, the means for determining filter coefficients is configured for determining the filter coefficients in dependence on a uniform power delay profile of the communi- cation channel. For example, the means for determining fil¬ ter coefficients is configured for receiving a control sig¬ nal from a means for providing channel information, the control signal indicating a maximum channel delay or a length of a power delay profile associated with a communi- cation channel. Based on this information, the means for determining filter coefficients may be configured for de¬ termining the filter coefficients such that the resulting pass-band has a width being e.g. approximately equal to a length of the power delay profile. This is due to the fact that a spectrum of a frequency domain signal resulting from time-frequency transforming a time domain signal is closely related to the time domain signal. Preferably, the means for determining filter coefficients is configured for matching the width of the pass-band of the low-pass filter to the channel delay for an efficient low-complexity chan¬ nel estimation. For example, if only certain sub-carriers are modulated by pilot symbols for channel estimation so that the sub- carriers are spaced apart by a number of sub-carriers, which are not modulated by pilot symbols, then the means for determining may be configured for determining the fil¬ ter coefficients from the following equations:

where

is a MxM autocorrelation matrix, No is a noise power and I denotes an identity matrix, and

is a IxM cross-correlation vector, M denoting a filter or¬ der, wherein a m-th row and a n-th column of i?_## is given by

fe_M}_m>n=E^_n)OH%_m)_Df|=R_m[(m-n)D_fl

wherein a m-th row of R_n^ (row vector) is:

wherein

wherein i is an index countin pilots only and i is a sub- carrier index for data and pilots, wherein

With these definitions the entries of the equations become:

wherein T denotes a sample interval, Tw denotes a delay be¬ ing equal to or greater than a maximum channel delay, and wherein Rgg^~ denotes an inverse of an autocorrelation ma¬ trix of the set of values, i.e. the autocorrelation matrix of an estimate of the channel transfer function.

Since, due to the sine-functions, all entries are real valued, the above matrix W including filter coefficients is also real-valued. Moreover, a matrix inversion is to be computed only once which introduces a further complexity reduction.

In accordance with a further aspect of the present inven¬ tion, if the low-pass filter 107 is an interpolation filter for performing e.g. a linear or polynomial interpolation, than the means for determining filter coefficients may be configured for determining the filter coefficients in de¬ pendence on a number of values, for which the estimate of the channel transfer function is to be interpolated. This issue will be addressed later in detail.

Moreover, the means for determining filter coefficients may be configured for determining a control signal indicating a width of the pass-band being associated with the determined filter coefficients. In order to shift the spectrum of the transformed signal towards the determined pass-band, the transformer 101 may be configured, in response to the con¬ trol signal, for pre-processing the time domain signal as has been described above or for post-processing the trans- formed signal as has been described above for shifting the spectrum of the transformed signal into the pass-band of the low-pass filter in order to take a possible change of the pass-band into account, when new filter coefficients are determined.

The present invention further provides a multi-carrier re¬ ceiver comprising the filter apparatus as described above, the filter apparatus being configured for transforming a received time domain multi-carrier signal into a trans¬ formed signal representing a spectrum of the received time domain multi-carrier signal, and for filtering the spectrum of the received time domain multi-carrier signal, i.e. the transformed signal, in order to obtain a received frequency domain multi-carrier signal. In addition, the multi-carrier receiver comprises means for extracting information from the received frequency domain multi-carrier signal.

For example, the means for extracting information is con- figured for equalizing the transformed signal using an es¬ timate of the channel transfer function provided by the low-pass filter, when the low-pass filter is configured for channel estimation.

In addition, the functionality of the inventive filter ap¬ paratus for frequency domain filtering may be used for re¬ ducing a phase drift affecting the transformed signal, wherein the phase drift may result when a frame synchroni¬ zation error in time domain occurs.

For example, the means for extracting information may com¬ prise a detector for detecting the phase drift in the transformed signal and for controlling the transformer in order to control the pre-processing or post-processing for introducing a correction phase drift to the transformed signal for at least partly compensating the phase drift. When transmitting an OFDM modulated signal over a multi- path fading channel, the received signal will have unknown amplitude and phase variations. After OFDM demodulation the channel response is described by the channel transfer func- tion (CTF) for sub-carrier i, H₁ - H(f = i/τ), where l/T de¬ notes the sub-carrier spacing. A snapshot of the magnitude and phase of the observed CTF is shown in Fig. 14 illus¬ trating that the phase of the CTF, φ_± = arg(H.), experiences a phase drift, defined by

where Aφ_± =

accounts for the phase between sub- carriers i-1 and i. The phase drift defines the average change of phase between two adjacent sub-carriers. For many applications this phase drift should be as small as possi¬ ble.

Phase drift in frequency domain results e.g. from frame synchronization errors, i.e. from timing errors introducing a phase shift in spectral domain, the phase shift increas¬ ing e.g. linearly over frequency.

It should be noted that non-perfect timing synchronization will also result in a change of the phase drift E[A^₁]. In fact, if the timing offset, T_off, does not exceed the guard interval length, T_GI , minus the maximum delay of the chan¬ nel τ_n , so T_nff ≤ T_rr - r _v , there will be no loss in or- thogonality of the OFDM signal. However, T_off ≠ 0 will re- suit in a different phase drift S[A^₁], as is described in M. Hsieh and C. Wei, "Channel Estimation for OFDM Systems Based on Comb-Type Pilot Arrangement in Frequency Selective Fading Channels," IEEE Transactions on Consumer Electron^¬ ics, vol. AA, pp. 217-225, Feb. 1998.

The phase drift may degrade the performance if space fre¬ quency codes or differential modulation is used. Further¬ more, the phase drift also degrades the performance of polynomial interpolation algorithms, e.g. linear or spline interpolation.

Fig. 14 shows a phase and a magnitude of a snapshot of a channel transfer function (CTF) . Fig. 15 shows a corre¬ sponding time domain snapshot of a channel impulse response (CIR) .

The non-zero phase drift E[Aφ_d] is typical for any OFDM system. It is due to the structure of the channel impulse response (CIR) , which is the inverse Fourier Transform of the CTF. The CIR which generates the CTF in Fig. 14 is shown in Fig. 15. As a general property, the CIR is only non-zero within the range [θ, r_max], where τ_max denotes the maximum delay of the channel.

Since the CIR is related to the CTF by a Fourier Transform it may be viewed as the spectrum of the CTF.

A non-zero phase drift can degrade a performance of an OFDM system, for example in a case of a differential phase cod¬ ing. This is due to the fact that a phase drift introduces an additional phase term leading to phase errors when per¬ forming a differential phase demodulation, so that informa- tion detection errors occur.

Furthermore, the phase drift can introduce channel estima¬ tion errors when estimating the communication channel, for example when estimating the channel transfer function, in order to e.g. equalize the received multi-carrier signal in frequency domain.

In order to estimate e.g. the channel transfer function, an OFDM transmitter may introduce so-called pilot symbols be- ing known in the receiver for channel estimation. Usually, the pilot symbols are used for modulating sub-carriers of a OFDM signal to be transmitted. At a receiver, the pilot symbols are removed from the modulated sub-carriers by the means of demodulation, e.g. by dividing the modulated sub- carriers by corresponding pilot symbols in order to obtain sub-carrier values comprising information with respect to the channel transfer function.

In Fig. 16, a block diagram of pilot symbol based channel estimation for OFDM is depicted. After transforming the re¬ ceived signal to frequency domain by the means of the Fast Fourier Transform (FFT) a transformed signal is obtained having values associated with sub-carriers, wherein only certain sub-carriers are modulated by pilot symbols for channel estimation. In order to extract the modulated sub- carriers, a de-multiplexer can be used in order to de¬ multiplex (DMUX) the modulated sub-carriers, which are, subsequently, provided to a channel estimator being config¬ ured for channel estimation. The channel estimator may per¬ form the pilot demodulation mentioned above in order to ex¬ tract the sub-carrier values, wherein, after having per¬ formed the de-modulation, the sub-carrier values are esti- mates of the channel transfer function at frequency points associated with sub-carriers modulated by the pilot sym¬ bols. In order to obtain a channel estimate for all sub- carriers, the channel estimator may, for example, be con¬ figured for performing an interpolation in order to provide channel estimates for all sub-carriers by means of interpo¬ lating between two subsequent channel estimates obtained from the modulated sub-carriers being spaced apart by a number of sub-carriers. In order to detect information con¬ tained by the transformed signal bypassed by the de- multiplexer, a detection unit (DT) is used. For example, the detection unit is configured for receiving the channel estimates in order to equalize the transformed signal. How¬ ever, if a phase drift occurs, then the channel estimates are erroneous due to an additional phase term. This leads to a performance degradation while detecting the informa¬ tion comprised by the transform signal after having equal¬ ized the transformed signal using the erroneous channel es¬ timates. In order to reduce the phase drift, a phase compensation can be performed, as is described e.g. in M. Hsieh and C. Wei, "Channel Estimation for OFDM Systems Based on Comb- Type Pilot Arrangement in Frequency Selective Fading Chan¬ nels," IEEE Transactions on Consumer Electronics, vol. 44, pp. 217-225, Feb. 1998, for the case of channel estimation. In particular, after a DFT and after a subsequent pilot signal extraction, a least squares estimate of pilot sig- nals is performed in order to provide an estimated channel transfer function. In a next step, a change in phase caused by frame error is determined from subsequent values of the channel transfer function. In a next step, the estimated change in phase is removed from the estimated channel transfer function and the resulting channel transfer func¬ tion is provided to a minimum mean squared error estimator for further channel estimation. After the MMSE channel es¬ timation, the channel transfer functions of data carriers are interpolated using linear or higher order interpola- tions. In a next step, a phase post-compensation is per¬ formed where the previously removed phase change is re¬ stored in order to provide an interpolated channel transfer function comprising the change in phase.

For differentially modulated signals, S. M. Weinfurtner, "Frequency-Domain Frame Synchronization for Optimum Fre¬ quency-Differential De-modulation of OFDM," in Proc. IEEE Global Telecommunications Conference (GLOBECOM '99), Rio de Janeiro, Brazil, pages 857-862, 1999 discloses applying a phase correction coefficient in the differential demodula¬ tion process in order to perform a phase rotation on sub- carrier amplitudes in frequency domain for compensating frame synchronization errors.

Further embodiments of the present invention will be de¬ scribed with reference to Figs. 2 to 12. Fig. 2a shows an OFDM receiver with phase compensation af¬ ter Fourier Transform, wherein the receiver comprises an antenna 201 coupled to a means 203 for removing guard in- terval, the means 103 for removing the guard interval being coupled to a serial to parallel converter 205 (S/P) . The serial to parallel converter 205 has a plurality of outputs coupled to a plurality of inputs of a Fourier transformer 207, the Fourier transformer 207 being configured for per- forming a Fast Fourier Transform (FFT) . The Fourier trans¬ former 207 has a plurality of outputs coupled to a phase compensator 209, the phase compensator 209 having a plural¬ ity of inputs being coupled to a detector not shown in Fig. 2a. It is to be noted that the Fourier transformer 207 and the phase compensator 209 are comprised by the inventive transformer mentioned above.

Fig. 2b shows a block diagram of an OFDM receiver having cyclic shift before performing the Fourier Transform by the Fourier transformer 207. The cyclic shift is performed by a cyclic shift element 301 coupled between the means 203 for removing the guard interval and the serial to parallel con¬ verter 205. In this embodiment, the cyclic shift element 301, the serial to parallel converter 205 and the Fourier transformer 207 constitute the inventive transformer, wherein the plurality of outputs of the transformer 207 is coupled to the detector, which is not shown in Fig. 2b. Furthermore, it is not shown in Figs. 2a and 2b that the detector may be configured for controlling the phase com- pensator 209 and the cyclic shift element 301.

As is depicted in Figs. 2a and 2b, the solutions shown therein are applicable to a wide range of OFDM receivers due to the OFDM standard conform structure. Moreover, if the phase compensation is applied according to the embodi^¬ ment of Fig. 2a, there is no need to compensate the induced phase shift after channel estimation and interpolation, contrary to the prior art approaches mentioned above. In particular, the negative effects of a non-zero phase drift from (1) can be compensated by a phase compensation unit after the FFT at the OFDM receiver, as is shown in Figs. 2a and 2b.

Another possibility to solve the problem related to the one-sided spectrum of the frequency response is to match the filter response to the observed characteristics of the OFDM signal. This is indicated in Fig. 5, where the pass- band of the channel estimation filter is chosen within the range [θ, τ_aax]. However, this will always result in complex valued filter coefficients.

In order to obtain a channel estimate with a filter having real valued coefficients, the inventive approach described above can be applied, wherein cyclic shift at the receiver or phase compensation at the receiver can be applied. Cy¬ clic shifts in the receiver are known from M. I. Rahman, K. Witrisal, D. Prasad, O. Olsen, and R. Prasad, "Performance Comparison between MRC Receiver Diversity and Cyclic Delay Diversity in OFDM WLAN Systems, "in Proc. Int. Symposium on Wireless Personal Multimedia Communications (WPMC 03), Yo- kosuka, Japan, Oct. 2003 and from A. Dammann and S. Kaiser, "Standard Conformable Antenna Diversity Techniques for OFDM and its Application to the DVB-T System," in Proc. IEEE Global Telecommunications Conference (GLOBECOM 2001), San Antonio, TX, USA, pages 3100-3105, Nov. 2001, wherein cy¬ clic shift is applied on the receiver side in order to ex- ploit spatial diversity. In accordance with the present in¬ vention, however, cyclic delays are not inserted in order to exploit spatial diversity, but in order to reduce phase drift or to obtain filter having real valued filter coeffi¬ cients.

We propose to cyclically shift the received signal before the FFT, as shown in Fig. 2b. Cyclically shifting the re- ceived signal by - δ_cyc samples before the FFT results in the following phase shift after the FFT:

βc_yc = 2*tf_cyc/AT_Fra ( 2 )

Fig. 3a shows a phase of a channel transfer function (CTF) after cyclically shifting a time domain signal before FFT. Fig. 3b shows a corresponding magnitude of the channel transfer function.

The effect of the cyclic shift to the received signal is Y₁ = Y^⁶"^ . If S_cyc is chosen according to (2), the effect to the received signal is identical, regardless whether a phase compensation after the FFT (Fig. 2a) or a cyclic shift before the FFT (Fig. 2b) is chosen.

In contrast to the cyclic shift operation as proposed in Fig. 2b, a phase shift according to Fig. 2a is more compu¬ tationally complex. While a cyclic shift can be performed very efficiently using a shift register of — δ samples, a phase compensation of θ_cyc degree per sub-carrier requires one multiplication by e^{j cyc±} per sub-carrier. On the other hand, given the overall complexity of an OFDM receiver, one additional multiplication per sub-carrier may not be that significant. It should be noted however, that the phase compensation as shown in Fig. 2 has not yet been proposed for all applications described in the remainder of this section.

Note, the cyclic shift only changes the phase of the CTF, the magnitude remains unaffected. This can be checked by comparing the CTF of an OFDM signal without and with cyclic shift shown in Figs. 14, 3a and 3b. Since the effects of the frequency selective channel are compensated by the channel estimator anyway, no other operations are neces¬ sary. A snapshot of the magnitude and phase of the CTF and the corresponding CIR after cyclically shifting the received signal is shown in Figs. 3a, 3b and Fig. 4, respectively. While there are still strong variations in amplitude and phase due to frequency selective fading, the phase drift E[Aq)₁] is compensated. The effective CIR of Fig. 4 is shifted towards negative delays. Instead of a one-sided spectrum, the received signal now has a two-sided spectrum.

The cyclic shift at the receiver side can also be applied to channel estimation based on the discrete cosine trans¬ form (DCT) . Since the DCT operates on a two-sided spectrum, the cyclic shift before OFDM demodulation may be very bene¬ ficial.

For pilot-symbol aided channel estimation (PACE) known sym¬ bols (pilots) are inserted, with an equidistant spacing of D_f sub-carriers. In order to reconstruct the signal, fil¬ tering and interpolation between the pilots is necessary. For channel estimation the received signal After OFDM de¬ modulation at the pilot positions, H₁ = H_j0 + Ny_D , with i = are used. An FIR filter with dimension M_f can be written in the general form

H₁ = H_DtI+Ai = ^M∑ W_m ■ H₂__m, i = D_fϊ + M ( 3 ) m=0

If the symmetries of the spectrum of W_m are such that the real and imaginary parts are an even and odd function, re¬ spectively, the coefficients of FIR interpolation and/or smoothing filters will be real valued. In general a real valued filter will only have half the computational cost of a complex valued filter.

In P. Hδher et al, ,,Pilot-Symbol-Aided Channel Estimation in Time and Frequency^λλ, in Proc. Communication Theory Mini- Conference (CTMC) within IEEE Global Telecommunications Converence (Globecom 97), Phoenix, AZ, USA, pp.90-96, 1997, PACE by Wiener filtering was proposed. In accordance with a further aspect of the present inven¬ tion, if a Wiener interpolation filter (WIF) with model mismatch is chosen, the filter W = ψ_xr"'_rW_M | is designed such that it covers a great variety of power delay pro¬ files. Accordingly, a rectangular shaped power delay pro¬ file with maximum delay T_N fulfils this requirement. The Fourier Transform of a uniform power delay profile, which is non-zero within the range [θ, T₁₇], yields the frequency correlation

_{[Ai] =} LΞM&M^I . _e-^Ψ _{( 4}, τtT_aΔ.l

Due to the complex phase term in R_m[Aϊ] the corresponding Wiener filter will also have complex coefficients.

However, a complexity of a filtering operation or of a channel estimation operation when using filter having com¬ plex valued coefficients is insignificantly increased when compared with filtering or channel estimation using filters having real valued coefficients only.

Suppose the CIR without cyclic shift is within the range [O'^_max] then a cyclic shift of - δ_cγc, - S_cyc denoting e left shift, will result in an effective CIR which is non-zero within the range [- δ_cyc, τ_max ~ S_cyc]. If the cyclic shift is chosen such that

T - A "eye ₂ w;

the effective CIR will now be non-zero only within the passband of the channel estimation filter. Given a filter with low-pass characteristics, having a symmetric two-sided passband within the range [-T_w/2,T_w/2], the effective CIR of the cyclically shifted signal will pass through the filter undistorted, as illustrated in Fig. 4. If the maximum delay of the channel T_n^_x can be estimated, T_w could be chosen according to T_w =r_max-fA_w, where A_w accounts for a roll-off delay, which may be inserted to non-perfect slope of the filter response. On the other hand, if r_max is not known it can be upper bounded by T_GI , so T_a = T_GI .

Now we can use a real valued frequency correlation function to generate the filter coefficients

Therefore, the WIF matched to the uniform power delay pro¬ file

is also real valued. This means that the computational cost is cut by half.

Alternatively to a mismatched WIF any FIR low pass interpo¬ lation filter benefits from the proposed cyclic shift. As long as the filter is matched to a passband of

the filter will be real valued. For such a filter, the per- formance will be optimum if the signal to be filtered passes through the filter unchanged.

Fig. 5 shows an OFDM receiver with phase compensation after the FFT in accordance with a further embodiment of the pre- sent invention.

Unlike the embodiment of Fig. 2a, the OFDM receiver shown in Fig. 5 comprises a phase compensator 501 having a plu¬ rality of outputs coupled to a detector 503, the detector 503 having a control output 505 coupled back to a control input of the phase compensator for providing information on the detected phase drift ©_cyc.

Fig. 6 shows an OFDM receiver with cyclic shift before the FFT. Unlike the embodiment shown in Fig. 2b, the OFDM re¬ ceiver shown in Fig. 6 comprises a cyclic shift element 601 coupled between the means 203 for removing guard interval and the serial to parallel converter 205, and a detector 603 coupled to the plurality of outputs of the Fourier transformer 207, wherein the detector 603 comprises a con¬ trol output 605 coupled back to a control input of the cy¬ clic shift element 601 for providing information on the number of values the time domain signal provided by the means 203 for removing the guard interval is to be shifted by.

Dependent on the application the cyclic shift may be chosen to compensate the phase drift of (1) , that is

OFDM systems with differential modulation or space- frequency coded OFDM systems will have an improved perform¬ ance if E[A(P₁] is estimated sufficiently well.

In order to integrate this phase drift compensation in the proposed receiver structure, an adaptive implementation ap¬ pears attractive, as shown in Fig. 5. For the proposed re¬ ceiver performing a cyclic shift before the FFT, this solu¬ tion can be implemented as shown in Fig. 6. Initially, the cyclic shift, δ_cyc may be set to a default value within the range [θ, T_βI/2]. Then, δ_cyc can be estimated and fed back to the cyclic shifting unit. Note, the phase drift is expected to change only on a long term basis, so no frequent updates are necessary.

In the following, a performance of the inventive approach for space-frequency block codes (SFBC) will be described.

In order to generate BER curves, an OFDM system using a space-frequency block code (SFBC) with N_τ = 2 transmit an- tennas has been implemented. An OFDM system with N_τ = 2 transmit and N_R = 1 receive antennas is used. The system parameters of the OFDM system and of the channel model are shown in Fig. 7. The channel is modelled by a tap delay line model with Q₀ = 12 taps, a tap spacing of Δr = 16 • T_spl , with an exponential decaying power delay pro¬ file, illustrated in Fig. 8. The total transmit power of the system is fixed, such that the total transmit power of a N₇. antenna system is equivalent to a single antenna sys¬ tem. No outer channel coding has been employed.

It is seen that for the considered system the BER floor can be somewhat reduced by optimizing the cyclic shift δ_cyc . For the considered parameters the optimum cyclic shift is about 44T_spI, which result in the best performance. It can also be seen that the accuracy of δ_cyc does not need to be high.

In accordance with the present invention, a performance of a polynomial interpolator can significantly be optimized. For PACE interpolation in frequency and time direction is necessary. While in time direction the Doppler power spec¬ trum has in general a symmetric two-sided and real valued profile (at least approximately) , the power delay profile is real valued but one-sided. In the following, the benefit of inserting a cyclic shift, δ will be described for a linear interpolator. The results are applicable for higher order polynomial interpolators as well.

Fig. 9 shows BER vs E_b/N₀ for N_τ = 2, and various cyclic shifts δ_cyc .

In accordance with the present invention, a performance of a polynomial interpolator can significantly be optimised. For linear interpolation, two successive pilot sub-carriers are used to determine the channel response for sub-carrier located in between these two pilots. For sub-carrier i, the channel estimate is given by

It is instructive to describe linear interpolation by a FIR filter from (3) . A linear interpolator can be described by the filter response

)

Inserting i^^iiΛ] into- FIR filter equation from (3) will ac- complish linear interpolation. The distortion imposed by the linear interpolator can be assessed by considering the spectrum of (8), i.e. the inverse Fourier Transform of wl^liπ], given by

_wllin](τ)

The spectrum of the linear interpolator is plotted in Fig. 10. First of all, a higher over-sampling rate compared to an ideal low-pass filter will be required. Second, for the best performance the spectrum of the signal to be in¬ terpolated should be concentrated around τ = 0 Since, the CIR is zero only within the range [θ_Λr_max], a cyclic shift of δ_cyc =

according to (7), will improve the interpo¬ lation performance.

Fig. 10 shows a time domain channel impulse response snap shot after cyclically shifting the signal before the FFT. In Fig. 10, also a frequency response of a linear interpo¬ lator is shown.

Fig. 11 shows MSE vs SNR for polynomial interpolation algo¬ rithms with a cyclic shift of S₁₀₁₀=O (dashed lines) and ^=44 (solid lines) . In the following, a performance of the inventive approach for some polynomial interpolation algorithms will be de¬ scribed. In Figure 13 the MSE is plotted against the SNR for various polynomial interpolation algorithms. It is seen that the linear interpolator with cyclic shift has a sig¬ nificantly lower error floor. The same is true for the spline interpolator, which gets close to the optimum Wiener filter if a cyclic shift is inserted.

In the following, further advantages associated with the inventive concept will be indicated.

Possible applications of the proposed cyclic shift at the OFDM receiver are e.g.:

• Providing real valued filter coefficients for PACE. This will cut the number of required multiplication per channel estimate by a factor of two, with identi¬ cal performance.

• Improving the performance of polynomial interpolation algorithms. The error floor caused by the interpola¬ tion error of polynomial interpolation algorithms can be improved significantly.

• Improving performance of differentially modulated sig¬ nals and space frequency codes.

Compared to inserting a phase shift, which requires one complex multiplication per sub-carrier, a cyclic shift op- eration is significantly easier to implement. Furthermore, no post-processing after channel estimation and/or demodu¬ lation is required. For instance, the solution proposed in M. Hsieh and C. Wei, "Channel Estimation for OFDM Systems Based on Comb-Type Pilot Arrangement in Frequency Selective Fading Channels," IEEE Transactions on Consumer Electron¬ ics, vol. 44, pp. 217-225, Feb. 1998.requires an additional multiplication per sub-carrier after interpolation. In the following, orthogonal frequency division multiplex¬ ing (OFDM) will be described.

For OFDM the signal stream is divided into N_c parallel sub-streams, typically for any multi-carrier modulation scheme. The 1^th sub-stream, commonly termed sub-carrier, of the £^th symbol block, named OFDM symbol, is denoted by X₁₁. An inverse DFT with N _FFT points is performed on each block, and subsequently the guard interval having N_GI sam¬ ples is inserted to obtain x_trD . After D/A conversion, the signal x(t) is transmitted over a mobile radio channel with response h(t, τ) . Assuming perfect synchronization, the re¬ ceived signal of the equivalent baseband system at sampling instants t = [n + £N_sym]r_spl is in the form

y_t,_B

( 10 )

where n(t) represents additive white Gaussian noise, and N_sym = N_FFT + N_GI accounts for the number of samples per OFDM symbol. At the receiver, the guard interval is removed and the information is recovered by performing a DFT on the re- ceived block of signal samples, to obtain the output of the OFDM demodulation Y_lΛ . The received signal after OFDM de¬ modulation is given by

where X_lΛ and H_tΛ denotes the transmitted information sym¬ bol and the channel transfer function (CTF) at sub-carrier i of the I^th OFDM symbol, respectively. The term N_{ri ac¬ counts for additive white Gaussian noise (AWGN) with zero mean and variance N₀. It is assumed that the transmitted signal consists of Jr OFDM symbols, each having N_c sub- carriers. We focus on channel estimation in frequency direction (sub- carrier index i) . Thus, the index denoting the OFDM symbol, £, will be dropped in the following.

We consider a time-variant frequency selective fading chan¬ nel, modeled by a tapped delay line with Q₀ non-zero taps. It is generally assumed that the channel is time limited by the maximum delay τ_mgx . Then the channel impulse response (CIR) h(t, τ) is zero outside the range [θ, τ_max] . It is com- monly assumed that the channel impulse response (CIR) is approximately constant during one OFDM symbol, so the time dependency of the CIR within one OFDM symbol can be dropped, i.e. h(t, τ) « h(τ) .

The channel transfer function (CTF) of (11), is the Fourier Transform of the CIR h^(A)(r) . Sampling the result at fre¬ quency f — ifT , the CTF at sub-carrier i is defined by H_J. = H(f = i/τ), where H(f) accounts for the analog CTF, and ^τ _syπ, = OW + ^N _GiK_Pi and T = N_FFTT_spl represents the OFDM sym- bol duration with and without the guard interval, respec¬ tively.

If the guard interval is longer than the maximum delay of the channel, i.e. T_GI ≥ τ_max , the orthogonality at the re- ceiver after OFDM demodulation is maintained, and the re¬ ceived signal of (11) is obtained.

In the following, an OFDM receiver with a cyclic shift at the receiver side will be described.

In this section the effective channel model after the cy¬ clic shift operation before the FFT. After the cyclic pre¬ fix is removed the received signal from (10) is cyclically delayed by — S_cyc samples

y_n ^{[ yc]}

JT h(t - T) ^■ x(τ)dτ + n(t) \_t=[n+s^hpl (12)

It is instructive to define an effective CIR, which de¬ scribes the equivalent CIR of the cyclically shifted signal

h(τ + δ_cycT_spl) (13)

While the maximum delay of the channel τ_max is not changed, the effective CIR, h^[cyό](τ) , is now non-zero within the range [- δ_cyc,τ_max - δ_cyc].

The FFT translates a cyclic delay into phase shifts. The effective CTF of the cyclic receiver is described by

2j[oyc] _ jj _eJ2πxδ_cyc/N_FBT ( 14 )

Now the cyclically shifted received signal after OFDM de¬ modulation can be expressed as

Y₁ = X₁E**⁰* + N₁ = X_jH_j.e'²'**'"'!""* + N₁ (15)

In the following, the principles of pilot-symbol aided channel estimation for OFDM will be described

For pilot-symbol aided channel estimation (PACE) , known symbols (pilots) are multiplexed into the data stream, which are used as side information to estimate the channel. PACE was first introduced for single carrier systems and required a flat-fading channel. To describe pilot symbol- assisted channel estimation it is useful to define a subset of the received signal sequence containing only the pilots, {xi^} = {x^}_/ with i = ±D_f . So, the pilot sequence is trans¬ mitted at a D_f times lower rate i

in frequency di¬ rection. (As a general convention, variables describing pi- lot symbols will be marked with a ~ in the following.) It is assumed that the pilots X_j are chosen from a PSK con¬ stellation, so Xr = 1. After OFDM demodulation, the received signal Y_x of (11) is obtained. For channel estimation the received signal at the pilot positions are de-multiplexed from the data stream, to obtain the received pilot sequence

Y- ^ X₁H₁ + N₂ = X_xH_x + N_x, with {i} e G ( 16)

where G is the subset of the OFDM frame containing the pi¬ lots.

In the following, OFDM channel estimation by FIR filtering will be described.

The first step in the channel estimation process is to re- move the modulation of the pilot symbols, which provides an initial estimate of the CTF at pilot positions

H- = X~Y~ = H- + X-N- (IV)

with X~X~ = 1

The channel estimator uses the demodulated pilots H- from (17) to yield the channel estimate

N_f-I H_x = ∑W_m • H-__e (18) m=0 where M_f denotes the filter order, i.e. the number of co¬ efficients of the FIR filter W_m .

The FIR filter W = [W₀, •••, W_M ^ may be implemented as e.g. low-pass interpolation filters, polynomial interpolators, or Wiener interpolation filters. The Wiener interpolation filter minimizes the mean squared error (MSE) between the desired response H_x and the observation, i.e. the received pilot symbols. This means that knowledge about the channel statistics is required. In contrast, low-pass interpolation filters and polynomial interpolators do not assume any knowledge of the channel statistics. For multi-carrier systems the observed channel is typically correlated in two dimensions, frequency and time. Moreover, the extension to PACE in two dimensions is possible.

In the following, Wiener filtering will be described.

The Wiener interpolation filter (WIF) is implemented by a FIR filter with M_f taps, according to (18) . The ^'WIF, W, is obtained by solving the Wiener-Hopf equation

In order to generate WIF, knowledge of the auto-correlation matrix and the cross-correlation vector are required

The entry of the m^th row and n^th column of the auto¬ correlation matrix of the CTF at pilot positions is given by

{R_M}_mtn=Efi_{T__n)DfH%_m)Df]=R_m[(™-")D_fl (21)

Provided the channel can be described by a tap delay line, the frequency correlation,

~ ⁿ)^DΛ between sub- carriers spaced Af = Ai]T Hz apart becomes

The m^th entry of the cross-correlation vector

can be expressed as

[R' JAi]) = E[H₁Hl ] = Ek₁Ht , I

i ~ D_fi In the following, the inventive mismatched estimator will be described

For the WIF the auto and cross-correlation functions need to he estimated at the receiver. It may be prohibitive to estimate the filter coefficients during operation in real time. Alternatively, a robust estimator with a model mis- match may be chosen. The filter W is designed such that it covers a great variety of power delay profiles. For exam¬ ple, a rectangular shaped power delay profile with maximum delay T_w fulfils this requirement. This assumption provides the frequency correlation function of the mismatched esti- mator, 2?j^^c)[Δi], from (6) . The mismatched estimator is de¬ termined by substituting i^^yc)[Δi] from (6) into (21) and (23) . Then the Wiener-Hopf equation (19) needs to be deter¬ mined only once. By using a mismatched estimator the filter coefficients can be pre-computed and stored.

It is important to note that the parameters of the robust estimator should always be equal or larger than the worst case channel conditions, i.e. largest propagation delays and maximum expected velocity of the mobile user. Further- more, the average SNR at the filter input, γ_w , which is used to generate the filter coefficients, should be equal or larger than actual average SNR, so γ_w ≥ γ_c . In order to determine the channel estimator only τ_a , F_w , and γ_w are required. If the maximum delay of the channel τ_max is not known it can be upper bounded by the guard interval dura¬ tion T₃₁. Since the filter should also satisfy the sampling theorem, the filter pass-band can be chosen within the range

In the following, estimating the phase drift will be de¬ scribed The phase drift, £[Δ<pJ, specifies the average change in phase between two adjacent sub-carriers, as defined in (1) . One possibility to estimate the phase drift is

Aφ_{x *} E[Aφ_±] (25)

which provides a sufficiently accurate estimate. More so¬ phisticated algorithms to estimate E[Aφ_d] are described in S. Kay, "A Fast and Accurate Single Frequency Estimator", IEEE Transactions in Acoustics, Speech and Signal Process¬ ing", vol. 37, pp. 1987-1990, December 1989.

Obviously, the CTF H₁ is not available at the receiver. Instead a noisy estimate of H₁ can easily be generated by H₁ « Y_±/Xi i where X₁ is a hard decision of X₁. The accuracy of (25) will degrade due to noise and decision feedback ef¬ fects. Fortunately, Aφ₁ does not need to be particularly accurate.

If pilot symbols are transmitted, e.g. for channel estima¬ tion, the phase drift may be estimated by

^AΦi

m-u « E[AφJ ( 26)

where H~ was defined in (17) . The division through D_f be¬ comes necessary, since the pilots are spaced D_f sub- carriers apart, i.e. D_f times the phase drift is esti¬ mated.

Fig. 12 shows a further embodiment of an OFDM receiver in accordance with the present invention.

Unlike the embodiment of Fig. 2a, the OFDM receiver shown in Fig. 12 comprises a de-multiplexer 1201 for de¬ multiplexing sub-carriers being modulated by pilot symbols for channel estimation. Optionally, the de-multiplexer 1201 (DMUX pilots) is configured for demodulating the sub- carriers being modulated by the pilot symbols. The de¬ multiplexer has an output coupled to the inventive appara- tus 1203 for reducing the phase drift. The apparatus 1203 for reducing the phase drift is coupled to a channel esti¬ mator 1205 being configured for estimating the channel transfer function in frequency domain. The channel estima¬ tor is coupled to means 1207 being configured for introduc- ing back the compensated phase shift so that all channel influences may be taken into account. The means 1207 for phase post-compensation is coupled to a means 1209 for ex¬ tracting information comprised by a transformed signal pro¬ vided by the FFT 207. In other words, the means 1209 for extracting information is configured for determining an in¬ formation amount comprised by a signal provided by the de¬ multiplexer 1201.

In accordance with a further aspect of the present inven- tion, the apparatus 1203 for reducing the phase drift (phase drift compensator) may be configured for providing a signal containing information on the phase drift to the means 1207 for phase post-compensation, so that in response to the signal provided by the apparatus 1203, phase post- compensation can be performed.

Moreover, the present invention provides concepts for fil¬ tering and interpolation for OFDM. Generally, the frequency response of the received signal after OFDM demodulation has a one-sided spectrum. By cyclically shifting the received signal before the FFT, the one-sided spectrum can be trans¬ formed into a symmetric two-sided spectrum. In general, the performance of standard interpolation algorithms such as linear or spline interpolation can be improved if the cy- clic shift is appropriately chosen. Furthermore, the per¬ formance of space frequency codes and differential modula¬ tion can also be improved. If the symmetries of the two- sided spectrum are such that the real and imaginary parts are an even and odd function, respectively, the coeffi¬ cients of FIR interpolation and/or smoothing filters become real valued, which cuts the computational cost by half.

Moreover, depending on certain implementation requirements of the inventive methods, the inventive methods can be im¬ plemented in hardware or in software. The implementation can be performed using a digital storage medium, in par¬ ticular a disk or a CD having electronically readable con- trol signals stored thereon, which can cooperate with a programmable computer system such that the inventive meth¬ ods are performed. Generally, the present invention is, therefore, a computer program product with a program code stored on a machine-readable carrier, the program code be- ing configured for performing at least one of the inventive methods, when the computer program products runs on a com¬ puter. In other words, the inventive methods are, there¬ fore, a computer program having a program code for perform¬ ing the inventive methods, when the computer program runs on a computer.

Claims

1. Filter apparatus for frequency domain filtering, the filter apparatus comprising:

a transformer (101) for time-frequency transforming a time domain signal into a transformed signal in fre¬ quency domain_/

a low-pass filter (107) for filtering the transformed signal in frequency domain, the low-pass filter (107) comprising real-valued filter coefficients represent¬ ing a real part of a filter response, wherein filter coefficients representing an imaginary part of the filter response are set to zero;

wherein the transformer (101) is configured for pre¬ processing the time domain signal in time domain be¬ fore time-frequency transforming or for post- processing the transformed signal in frequency domain prior to filtering in order to introduce a phase shift to the transformed signal for shifting a spectrum of the transformed signal towards a pass-band of the low- pass filter (107) .

2. Apparatus according to claim 1, wherein the trans¬ former (101) is configured for pre-processing the time domain signal or for post-processing the transformed signal in order to shift the spectrum of the trans- formed signal such the shifted spectrum of the trans¬ formed signal and the pass-band of the low-pass filter (107) overlap.

3. Filter apparatus according to claim 1 or 2, wherein the time domain signal is a received version of a transmit signal, the transmit signal being transmitted through a communication channel, wherein a width of the pass-band of the low-pass filter (107) is upper- bounded by a maximum channel delay.

4. Apparatus according to claim 3, wherein the width of the pass-band of the low-pass filter (107) is deter¬ mined by a channel delay of a certain communication channel, the certain communication channel having a maximum channel delay among a plurality of communica¬ tion channels to be considered when receiving the transmit signal.

5. Filter apparatus according to claim 3 or 4, wherein the width of the pass-band is equal to or smaller than the maximum channel delay plus an additional delay in- troduced by a slope of a filter response of the low- pass filter (107) .

6. Filter apparatus according to claims 1 or 5, wherein the transformer (101) comprises a cyclic shift element for pre-processing the time domain signal, the cyclic shift element being configured for cyclically shifting the time domain signal by a number of values in order to introduce a phase shift to the transformed signal in frequency domain.

7. Filter apparatus according to claim 6, wherein the number of values is dependent on a width of the pass- band of the low-pass filter (107) .

8. Filter apparatus according to claim 7, wherein the number of values is equal to or smaller than half the difference between the width of the pass-band and an additional delay introduced by a slope of the filter response of the low-pass filter (107) .

9. Filter apparatus according to claims 6 to 8, wherein the cyclic shift element comprises a shift register, the shift register having an input and an output, the output being coupled to the input for cyclically shifting the time domain signal.

10. Filter apparatus according to claims 1 to 9, wherein the transformer (101) comprises a phase compensator for post-processing the transformed signal in fre¬ quency domain, the phase compensator being configured for changing a phase of the transformed signal in or¬ der to introduce the phase shift.

11. Filter apparatus according to claim 10, wherein the phase compensator is configured for phase shifting the values of the transformed signal by a phase shift, the phase shift being dependent on the width of the pass- band of the low-pass filter.

12. Filter apparatus according to claims 1 to 11, wherein the time domain signal is a received version of a transmit signal being transmitted through a communica- tion channel, wherein the low-pass filter (107) is configured for channel estimation.

13. Filter apparatus according to claim 12, wherein the transformed signal comprises a set of received ver- sions of sub-carrier values resulting from modulating sub-carrier values by pilot symbols for channel esti¬ mation, wherein the transformer (101) is further con¬ figured for demodulating the received versions of sub- carrier values using the pilot symbols in order to provide a set of values to the low-pass filter (107) for channel estimation, the low-pass filter (107) be¬ ing configured for estimating the channel on a basis of the set of values.

14. Filter apparatus according to claim 13, wherein a width Tw of the passband the low-pass filter (107) is within the following range:

wherein r_max denotes a maximum channel delay, T_GI a length of a guard interval, D_f a spacing between sub- carriers being modulated by pilot symbols and T a sym¬ bol duration

15. Filter apparatus according to claim 13 or 14, wherein the low-pass filter (107) comprises a Wiener filter for performing the channel estimation.

16. Filter apparatus according to claims 13 to 15, wherein the low-pass filter (107) is an interpolation filter.

17. Apparatus according to claims 1 to 16, further com¬ prising means for determining filter coefficients.

18. Apparatus according to claim 17, wherein the means for determining filter coefficients is configured for de- termining the filter coefficients in dependence on a uniform power delay profile of the communication chan¬ nel.

19. Filter apparatus according to claim 17 or 18, wherein the transformed signal comprises a set of received versions of sub-carrier values resulting from modulat¬ ing sub-carrier values by pilot symbols for channel estimation, wherein the transformer (101) is further configured from demodulating the received versions of sub-carrier values using the pilot symbols in order to provide a set of values, the set of values comprising channel estimates for sub-carriers being modulated by pilot symbols, the sub-carriers being modulated by pi¬ lot symbols being spaced apart by a number of sub- carriers, wherein the means for determining filter co¬ efficients is configured for determining the filter coefficients from the following equation: —\

W = R_HΑ -R_M

wherein R_Hfj denotes a real-valued cross-correlation vector comprising only real-valued coefficients being derivable from the following equation

wherein T denotes a sample interval, Tw denotes a de¬ lay being equal to or greater than a maximum channel delay, k is an entry index, and wherein Rjyf denotes an inverse of an autocorrelation matrix of the set of values.

20. Filter apparatus according to claims 17 to 19, wherein the means for determining filter coefficients is con- figured for determining a control signal indicating a width of the pass-band, and wherein the transformer (101) is configured, in response to the control sig¬ nal, for pre-processing the time domain signal or for post-processing the transformed signal for shifting the spectrum of the transformed signal into the pass- band of the low-pass filter (107.

21. Multi-carrier receiver, comprising:

the filter apparatus according to claims 1 to 19, the filter apparatus being configured for transforming a received time domain multi-carrier signal into a transformed signal representing a spectrum of the re¬ ceived time domain multi-carrier signal, and for fil- tering the transformed signal in order to obtain a re¬ ceived frequency domain multi-carrier signal; and means for extracting information from the received frequency domain multi-carrier signal.

22. Method for frequency-domain filtering, the method com- prising:

time-frequency transforming a time domain signal into a transformed signal in frequency domain;

low-pass filtering the transformed signal using a low- pass filter comprising real valued filter coefficients representing a real part of a filter response, wherein filter coefficients representing an imaginary part of the filter response are set to zero;

pre-processing the time domain signal before time- frequency transforming or post-processing the trans¬ formed signal before filtering in order to introduce a phase shift to the transformed signal for shifting a spectrum of the transformed signal towards a pass-band of the low-pass filter.

23. Method for receiving multi-carrier signals, the method comprising:

the method according to claim 22 for transforming a received time domain multi-carrier signal into a transformed signal representing a spectrum of the re¬ ceived time domain multi-carrier signal, and for fil- tering the transformed signal in order to obtain a re¬ ceived frequency domain multi-carrier signal; and

means for extracting information from the frequency domain multi-carrier signal.

24. Computer program for performing the method according to claim 22 or for performing the method according to claim 23, when the program runs on a computer.