Frequency Domain Frequency Shift for Optimal Filtering of Cyclostationary Signals

Digital Signal Processing 12, 561–589 (2002) doi:10.1006/dspr.2001.0410

Frequency Domain Frequency Shift for Optimal Filtering of Cyclostationary Signals Gareth Parker Defence Science and Technology Organisation, Salisbury, South Australia, Australia; and Institute for Telecommunications Research, University of South Australia, South Australia, Australia E-mail: [email protected] Parker, G., Frequency Domain Frequency Shift for Optimal Filtering of Cyclostationary Signals, Digital Signal Processing 12 (2002) 561–589. Optimum reconstruction of corrupted cyclostationary signals is achieved using the filter class known as the frequency shift filter. This filter requires the received signal to be shifted by the frequencies of cyclostationarity of the signal and with a frequency domain implementation it will often be best to effect the frequency shifts directly in the frequency domain. This paper introduces techniques for exactly achieving these shifts as well as providing more computationally efficient approximate solutions.  2002 Elsevier Science (USA)

1. INTRODUCTION The optimum 1 filter for reconstruction of a cyclostationary signal of interest (SOI), s(t), corrupted by channel impairments such as interference is the frequency shift (FRESH) filter [1]. This filter exploits the correlation exhibited between spectral components of s(t), separated by the frequencies of cyclostationarity and which may also exist between these components and similarly separated components of the conjugate s ∗ (t). In the time domain the FRESH filter is a periodically time varying filter, whereas in the frequency domain it may be interpreted as a multivariate Wiener filter with input signals comprising the received signal x(t), its conjugate x ∗ (t), and appropriately frequency shifted versions of these. In this paper we consider the frequency domain realisation. In practice, a suboptimum FRESH filter, combining a subset, N , of all possible frequency shifted x(t) and x ∗ (t), may be preferable to the optimum filter, although optimum performance may still be achieved if sufficient redundancy exists in the chosen inputs. One method of implementing the frequency domain 1 In this case, by optimum we mean the linear filter which minimises the mean squared signal reconstruction error in the presence of stationary noise.

561 1051-2004/02 $35.00  2002 Elsevier Science (USA) All rights reserved.

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FRESH filter is to perform the frequency shifts in the time domain via mixing with appropriate oscillators prior to the frequency domain transformation. This method requires N + 1 frequency domain transformers, including one for the reference input, and may be too computationally intensive in practice, particularly for large N . A better approach may be to use a single transformer for each of x(t) and x ∗ (t) and to perform the frequency shift within the frequency domain. An even better approach may be to use a single transformer for x(t) and to derive from this the transform of x ∗ (t) in addition to the required frequency shifts. The benefit of these approaches is particularly high if the number of frequency domain processors is limited, which may well be the case, especially if the processor is an expensive, high performance unit. ˆ ) = X(f )H(f ), where A FRESH filter estimates the SOI using the product S(f X(f ) = [X1 (f ), . . . , XN (f )] is the vector of frequency shifted signal Fourier transforms (possibly conjugated) and H(f ) = [H1 (f ), . . . , HN (f )]T is the vector of transfer functions which are applied to each frequency shifted signal. The transfer functions are given by H(f ) = R−1 (f )S(f ) where S(f ) = [Ssx1 (f ), . . . , SsxN (f )]T and



Sx1 x1 (f )  .. R(f ) =  . Sx1 xN (f )

...

 SxN x1 (f )  .. . .

(1)

. . . SxN xN (f )

Sxu xv (f ) is the cross spectrum between the frequency shifted inputs Xu (f ) and Xv (f ) and Ssxu (f ) is the cross spectrum between the reference S(f ) and Xu (f ). At a particular frequency fk Hz, cross spectra may be estimated by the cross correlation between the fk Hz frequency components of the two signals and in practice this will be performed over a time interval, t seconds. Ideally, the SOI components of the FRESH filter inputs will be perfectly correlated. However, this cannot occur unless the frequency shifts have been exactly effected and, as t → ∞, the correlation measured between inaccurately shifted frequency components decreases. A method is required to accurately achieve these frequency shifts. Inspection of the FRESH filter equation (1) suggests two desirable characteristics of an approximate frequency shift: (1) high correlation and (2) small mean squared error, between the approximately shifted frequency components and those which would be obtained using an exact frequency shift. In this paper, we derive a technique for effecting an exact frequency domain frequency shift as well as a number of more practical approximations to this. Through simulation results we compare the performance of these techniques both by directly measuring the FRESH filter reconstruction error and by examination of the desirable properties just discussed.

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FIG. 1. Analysis and synthesis filter banks.

2. EXACT FINE FREQUENCY SHIFT A frequency shift of − Hz may be applied to the discrete signal x(n) by time domain mixing with the complex oscillator ψ(n) = e−j 2πn/fs , where fs is the sampling frequency. We seek an operator g{·} to achieve this same result in the discrete frequency domain so that g{F {x(n)}} = F {e−j 2πn/fs x(n)}. Through application of this operator we say an exact frequency shift 2 has been achieved. We implement the transformation into the frequency domain, F {·}, using a K bin filter bank analyser [2] as shown in Fig. 1 together with the inverse transformer or synthesiser. In the figure ↓ M and ↑ M represent sampling rate decimation and expansion by M. The h(n) and f (n) are known as the analysis and synthesis filters. The output of the analysis filter bank at time mM/fs is a vector of bins X(m) = [X(m, f0 ), . . . , X(m, fK−1 )], the kth one containing an estimate of the complex envelope of the narrow bandpass filtered component of x(n) centred at fk Hz. This envelope is sometimes called a complex demodulate [3] and may be expressed as X(m, fk ) =

∞


h(n)x(mM − n)e−j 2π(mM−n)fk /fs .

(2)

n=−∞

We consider the bins to be uniformly spaced between −fs /2 and fs /2 (or equivalently 0 and fs ) and so the bin width is equal to fb = fs /K. The kth bin of 2 For our purposes an exact frequency shift is not necessarily one which does not corrupt the signal. The effect of aliasing must be considered when shifting in frequency any sampled signal and may result from either the time domain shift or the frequency domain application of g{·}.

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the transform of x(n) shifted in the time domain by − Hz is given by ∞


F {ψ(n)x(n)}k = =

h(n)x(mM − n)ψ(mM − n)e−j 2π(mM−n)fk /fs

n=−∞ ∞


h(n)x(mM − n)e−j 2π(mM−n)(fk +)/fs

n=−∞

= X(m, fk + )

(3)

which is the frequency component of x(n) centred at fk + . The vector of all such shifted bins corresponding to k = 0, . . . , K − 1 is denoted X (m). If  = fb , where  is an integer, the frequency shift is effected almost trivially using a circular rotation through the  filter bank bins [4]. However, this method is only exact when the required shift is a multiple of the bin width. In practice, this may not be the case and a residual fine shift will normally be required. We will generally assume that the coarse component of the frequency shift has been effected via bin rotation and that the  Hz shift achieves this residual fine shift. In practice the analysis filter has an impulse response h(n) with finite length Nh = RK and the summation in Eq. (2) is performed over T = Nh /fs s. In Section 2.1 we consider the simple case where R = 1 and in Section 2.2 we look at the more general and more useful case where R is an integer greater than 1. First we obtain an expression for X(m, fk ) in a form which is convenient for the analysis ahead. Consider an alternative representation of X(m, fk ) obtained by taking an exponential term outside the summation, which we now write with the finite limits, X(m, fk ) = e

−j 2πmMfk /fs

Nh


h(n)x(mM − n)ej 2πnfk /fs .

(4)

n=1

Next, consider the change of variables, r = Nh − n so that we have X(m, fk ) = e−j 2πmMfk /fs

N
h −1

h(Nh − r)x(mM − Nh + r)ej 2π(Nh −r)fk /fs .

(5)

r=0

Using the fact that Nh fk /fs = RKk/K is an integer, this becomes X(m, fk ) = e−j 2πmMfk /fs

N
h −1

h(Nh − r)x(mM − Nh + r)e−j 2πrfk /fs .

(6)

r=0

We can similarly write X(m, fk + ) = e

−j 2πmMfk /fs

N
h −1

h(Nh − r)x(mM − Nh + r)

r=0

× ψ(mM − Nh + r)e−j 2πrfk /fs .

(7)

Equations (6) and (7) express X(m, fk ) and X(m, fk + ) in a form which allows the discrete Fourier transform (DFT), or a computationally efficient fast Fourier transform (FFT), to be used to compute the summation.

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2.1. Condition Nh = K When the impulse response duration, Nh , equals the number of filter bank bins, K, the summation in (6) may be computed by direct application of a K point DFT. Define E(m) = [ej 2πmMf0 /fs , . . . , ej 2πmMfk /fs , . . . , ej 2πmMfK−1 /fs ] and denote the DFT computed on the block h(Nh − r)x(mM − Nh + r), r = 0, . . . , K − 1 by DFT{h(Nh )x(mM − Nh )}. Using an element-by-element vector middle product defined as [a1 , . . . , an ] ⊗ [b1 , . . . , bn ] = [a1 b1 , . . . , an bn ] we can write X(m) = E∗ (m) ⊗ DFT{h(Nh )x(mM − Nh )}

(8)

X (m) = E∗ (m) ⊗ DFT{h(Nh )x(mM − Nh )ψ(mM − Nh )}.

(9)

and

Using the relation DFT{ab} = (1/K)DFT{a} ∗ DFT{b} with ∗ denoting circular convolution and defining the DFT computed on ψ(mM − Nh + r) by DFT{ψ(mM − Nh )} we find X (m) =

1 ∗ E (m) ⊗ [DFT{h(Nh )x(mM − Nh )} ∗ DFT{ψ(mM − Nh )}]. K

(10)

As ψ(n) is a complex exponential, its DFT may be computed at time mM as its m = 0 DFT, "0 , mixed with a − Hz oscillator at the decimated sampling rate so that DFT{ψ(mM − Nh )} = e−j 2π(mM−Nh )/fs "0 . Such an update yields the expression X (m) =

1 −j 2π(mM−Nh )/fs ∗ e E (m) ⊗ [DFT{h(Nh )x(mM − Nh )} ∗ "0 ]. K

(11)

Using (8) we can rewrite DFT{h(Nh )x(mM − Nh )} = E(m) ⊗ X(m) and thus X (m) =

1 −j 2π(mM−Nh )/fs ∗ e E (m) ⊗ [(E(m) ⊗ X(m)) ∗ "0 ]. K

(12)

Equation (12) achieves the aim of effecting an exact frequency shift of x(n) by operating upon X(m). At first this appears to represent a somewhat computationally expensive process, but the complexity is dramatically reduced if a simple but accurate approximation is made. As the DFT of ψ(n) is that of a complex exponential it comprises relatively few significant frequency samples. The circular convolution may thus be replaced with a truncated linear convolution with little loss in precision. In the simplest approximation, "0 may be replaced by single sample complex scalar Kej φ and (12) reduces further to X (m) ≈ e−j 2πmM/fs +j φ X(m).

(13)

Under these conditions, the frequency shift operation is approximated simply by mixing the filter bank subband components with a − Hz oscillator operating at the decimated sampling rate. A similar technique to this subband mixing approach has been used for the estimation of the cyclic cross spectrum [3]

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and the author is aware of its use in practice for another application requiring frequency shifts.

2.2. Condition Nh > K The more general case where Nh > K will now be analysed. This case is particularly important and is usually preferred in all frequency domain signal reconstruction applications as the high resolution afforded by Nh = RK can have a large impact on the quality of signal reconstruction [5, 6]. Where Nh > K, time aliasing the Nh length block of h(Nh − r) x(mM − Nh + r) into a K sample block by stacking and adding may be used to manipulate the summation of (6) into a K point summation. The weighted overlap add (WOLA) [2] filter bank architecture implements the summation in (6) as the double sum N
h −1

h(Nh − r)x(mM − Nh + r)e−j 2πrfk /fs

r=0

=

K−1
R−1


h(Nh − r − lK)x(mM − Nh + r + lK)e−j 2πrfk /fs .

(14)

r=0 l=0

Maintaining the same nomenclature as in Section 2.1, the filterbank analyser output vector may now be expressed as ∗

X(m) = E (m) ⊗

 R−1


 DFT{h(Nh − lK)x(mM − Nh + lK)} .

(15)

l=0

The summation in (7) may be expanded in the same way so that N
h −1

h(Nh − r)x(mM − Nh + r)ψ(mM − Nh + r)e−j 2πrfk /fs

r=0

=

R−1
K−1
l=0 r=0

h(Nh − r − lK)x(mM − Nh + lK + r) × ψ(mM − Nh + lK + r)e−j 2πrfk /fs .

(16)

The vector of all shifted demodulates may then be expressed as  R−1 
X (m) = E∗ (m) ⊗ DFT{h(Nh − lK)x(mM − Nh + lK)ψ(mM − Nh + lK)} =

l=0  R−1


1 ∗ E (m) ⊗ K

DFT{h(Nh − lK)x(mM − Nh + lK)} 

l=0

∗ DFT{ψ(mM − Nh + lK)}  R−1
1 ∗ = E (m) ⊗
0 ∗ e−j 2π(mM−Nh +lK)/fs K l=0

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 × DFT{h(Nh − lK)x(mM − Nh + lK)} =

1 −j 2π(mM−Nh )/fs ∗ e E (m) K   R−1
−j 2πlK/fs e DFT{h(Nh − lK)x(mM − Nh + lK)} . ⊗ "0 ∗

(17)

l=0

Comparison of Eqs. (15) and (17) shows that in general, it is not possible to express X (m) in the form of a function of X(m). 3 However, two alternative methods of obtaining an exact frequency domain fine frequency shift are readily identified if the filter bank architecture is modified. First, Eq. (17) may be implemented directly using a modified WOLA architecture. In the conventional WOLA architecture, for the mth data block, R segments of windowed data are stacked and added, with a K point DFT performed on the sum to generate X(m). An equivalent operation is to reverse the order of summation and perform a K point DFT upon each of the R segments followed by a stack and add process. By multiplying each of these R DFTs by the appropriate exponential prior to summation, Eq. (17) is implemented. The disadvantage to this modified WOLA method is obvious: R × K length DFTs are required to compute each demodulate as opposed to a single DFT using the basic WOLA. A second alternative is to compute (3) using a K  = Nh = RK point DFT and use the same approach as with Nh = K to effect the fine frequency shift using Eq. (12). Following convolution, the K frequency domain samples are then obtained by decimating in frequency to retain only one in every R filter bank bins. The fine shifted filter bank bins may then be operated upon by the FRESH filter algorithm as if they were directly generated using a K bin filter bank. The synthesis process will be unchanged and will require only K bins. This second approach also requires R times as many computations as a direct K bin filter bank for the analysis DFT process, but additionally requires R times as many computations for the analysis filtering, which is applied to all K  analysis channels. In additon, the oscillator transform must also be computed over K  points, although this is a once-off operation and the overall computational expense is insignificant. This second approach may be preferable to the modified WOLA in instances where the use of a single FFT processor is essential or where similarity with the Nh = K method is desirable. This second method is also independent of the filter bank architecture as it operates on the subband components themselves. Hence, any architecture may be used for the filter bank, including both the WOLA and the polyphase filter approach [2], although the filter bank processor must support differing analyser and synthesiser resolution. 3 Unless the exponential multiplier within the summation of (17) reduces to a constant independent of l which will occur if  is a multiple of the bin width fs /K.

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3. APPROXIMATE FREQUENCY SHIFTS USING SUBBAND MIXING AND BIN COMBINING We have seen how an exact fine shift may be implemented by a combination of coarse bin rotations together with circular convolution with an oscillator DFT and that for Nh = K or R = 1, the simplest approximation to this is to truncate the oscillator transform to a single frequency sample. The resulting implementation is a mixing of the filter bank bins with a low sample rate oscillator. A similarly simple implementation may be alternatively derived for the general case of arbitrary R, using a frequency-shift interpretation of the shift filter bank analyser. Consider again Eq. (3) and redistribute the exponential components so that the analysis filter is associated with ej 2πn/fs X(m, fk + ) =

∞


h(n)x(mM − n)e−j 2π(mM−n)(fk +)/fs

n=−∞

= e−j 2πmM/fs

∞


h(n)ej 2πn/fs x(mM − n)e−j 2π(mM−n)fk /fs . (18)

n=−∞

Now, by letting

h (n) = h(n)ej 2πn/fs

X(m, fk + ) = e−j 2πmM/fs

this becomes ∞


h (n)x(mM − n)e−j 2π(mM−n)fk /fs .

(19)

n=−∞

Equation (19) represents processing the data with a DFT filter bank using analysis filters which are shifted in frequency by + Hz and then followed by a frequency domain mixing with a − Hz oscillator. Obviously, we would not want to explicitly affect such a shift of the analysis filter, since this would require that a separate filter bank be used for every different frequency shift. However, we can approximate the effect of the shifted analysis filter by a frequency domain operation on the signal demodulate. To derive an appropriate frequency domain operation, we consider approximations to h (n). First, consider crudely ˜ = h(n)ej φ . Substituting this into (19) gives us an approximating h (n) by h(n) approximation to the demodulate X(m, fk + ), ˜ X(m, fk + ) = e−j 2πmM/fs +j φ

∞


h(n)x(mM − n)e−j 2π(mM−n)fk /fs

n=−∞

=e

−j 2πmM/fs +j φ

X(m, fk ).

(20)

This is the kth bin of the conventional filter bank transform, mixed in the frequency domain with an − Hz oscillator. The frequency responses of X(fk ) ˜ k + ) are shown in Fig. 2 for the ideal case where H (f ) has an ideal and X(f bandpass form with bandwidth fs /K. From this figure, we can see that the ˜ frequency response of X(m, fk + ) extends from −fs /(2K) −  Hz to fs /(2K) −  Hz. This only partially overlaps the bandwidth −fs /(2K) Hz to fs /(2K) Hz ˜ fk + ) there are which is occupied by the exactly shifted X(m, fk + ). In X(m, some additional spectral components and there are some which are missing. ˜ fk + ) will be less than The crosscorrelation between X(m, fk + ) and X(m,

Gareth Parker: Optimal Filtering of Cyclostationary Signals

FIG. 2.

569

Fine frequency shift approximation using frequency shifted analysis filter.

the autocorrelation of X(m, fk + ) and the mean square error (MSE) measured between the two components will also be nonzero. The additional components present the lesser problem since these will be attenuated to some extent by the synthesis filter. However, the missing components are more problematic, particularly if  is large. The inclusion of the term φ ensures phase match ˜ in the region of spectral overlap. Since the group delay between h (n) and h(n) of the phase linear h(n) is (Nh − 1)/2, it follows that φ must be equal to φ = π(Nh − 1)/fs . The region of spectral overlap between the exact and approximately shifted demodulates can be extended to cover the full bandwidth of X(m, fk + ) by approximating h (n) by the sum of two filters, one unshifted and the other ˆ shifted by exactly one (possibly negative) bin width, i.e., use h(n) = h(n)(ej φ + j 2πn/K+j φ j φ j φ c ) or equivalently, H ˆ (f ) = H (f )e +H (f −fs /K)e c . Figure 3 depicts e ˆ k + ) this situation. In this case it can be clearly seen that the new estimate X(f contains frequency components which span the entire bandwidth −fs /(2K) ˆ k + ) will be close to to fs /(2K). The correlation between X(fk + ) and X(f the autocorrelation of X(fk + ). However, due to the excess bandwidth of Hˆ (f ), the amount of out-of-band noise will increase. This approach requires ˆ that the composite analysis filter, h(n) has a frequency response with close to unity gain and linear phase over the band of interest. The latter condition is achieved if h(n) has linear phase and if φc is chosen so that H (f − fs /K)ej φc matches the phase of H  (f ) over the region of spectral overlap. This requires φc = π(fs /K − )(Nh − 1)/fs or equivalently, φc = φ + Rπ . We will discuss in Section 3.1 how the former condition can be met by suitable choice of h(n). This approximation to the analysis filter h (n) may be easily implemented using the approach derived below. Consider a filter bank analyser which uses an ˆ analysis filter h(n) = h(n)ej φ (1 + ej 2πn/K+j Rπ ) and mix the decimated frequency

FIG. 3. Fine frequency shift approximation using extended frequency shifted analysis filter.

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domain samples with a low frequency complex oscillator so that ˆ X(m, fk + ) ∞


= e−j 2πmM/fs = e−j 2πmM/fs

ˆ h(n)x(mM − n)e−j 2π(mM−n)fk /fs

n=−∞ ∞


h(n)ej φ (1 + ej 2πn/K+j Rπ )x(mM − n)e−j 2π(mM−n)fk /fs

n=−∞

= e−j 2πmM/fs +j φ +e

∞


h(n)x(mM − n)e−j 2π(mM−n)fk /fs

n=−∞ ∞
−j 2πmM/fs +j φ

h(n)ej 2πn/K+j Rπ x(mM − n)e−j 2π(mM−n)fk /fs

n=−∞

˜ = X(m, fk + ) +e

−j 2πmM/fs +j φ+j Rπ

∞


h(n)x(mM − n)e−j 2π(mM−n)(fk +fs /K)/fs ej 2πmM/K

n=−∞

˜ = X(m, fk + ) +e

j 2πmM/K+j Rπ −j 2πmM/fs +j φ

e

∞


h(n)x(mM − n)e−j 2π(mM−n)(fk+1 )/fs

n=−∞

˜ ˜ = X(m, fk + ) + ej 2πmM/K+j Rπ X(m, fk+1 + ).

(21)

The approximation to the filter bank transform using h (n) may thus be effected by performing a filter bank transform upon x(n) using h(n), fine frequency shifting using subband oscillator mixing and combining adjacent bins. This bin combining appears to have the form similar to a convolution. Just as ˜ the approximation h(n) = h(n)ej φ gives rise to a similar solution to the simplest circular convolution approximation, this extension of the analysis filter gives rise to a convolution-like solution. Note that it can be shown that for a + Hz shift, ˆ ˜ ˜ X(m, fk − ) = X(m, fk − ) + e−j 2πmM/K−j Rπ X(m, fk−1 − )

(22)

˜ X(m, fk − ) = ej 2πmM/fs +j φ X(m, fk )

(23)

where

and φ = −π(Nh − 1)/fs . Figure 4 illustrates the process of frequency shifting via subband mixing and the advantage of bin combining. The top plot (a) shows the transform magnitude of a typical BPSK SOI, (b) shows the demodulate centered at fk Hz, and plot (c) shows the same demodulate shifted by exactly − Hz. Plot (d) shows the approximate shift obtained using subband mixing, together with the adjacent bin with which it may be combined. The spectral components which are common to the exact frequency shifted demodulate and those obtained via subband mixing and bin combining are shown in plots (e) and (f).

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FIG. 4. Depiction of exact and approximate fine shifted demodulate via subband oscillator mixing and bin combining.

3.1. Additional Criteria for the Design of Analysis and Synthesis Filters The design of analysis and synthesis filters for filter banks used in a backto-back arrangement where the input signal is filtered using a modification of the filter bank bins is a topic that has been extensively dealt with in the literature [2, 7]. There are usually two criteria that must be satisfied. The first, the perfect reconstruction criterion requires that a back-toback analysis–synthesis processor should, in the absence of frequency domain modification, perfectly reconstruct the input signal to within some designated tolerance. A tolerance that may satisfy most applications would be that the

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reconstruction error power is more than 40 dB below the signal power. The second criterion, the Parseval criterion requires that the summation of the power in the filter bank bins should equal the signal power measured in the time domain. The first criterion will be satisfied if the decimation factor M is chosen to prevent aliasing and the analysis and synthesis filters satisfy the property 1


H (f + ifs /K)F (f + ifs /K) = 1

for − fs /2 ≤ f ≤ fs /2

(24)

i=−1

provided their transition bands overlap only adjacent bins [2]. A simple way of satisfying this is to choose F (f ) so that it has unity gain over the pass and transition bands of H (f ) and then design H (f ) independently. The product H (f )F (f ) must have gain equal to −6 dB midway between filter bank bins (fs /2K) and under this design this requires H (f ) to have −6 dB gain at fs /2K. Filters based upon standard windows such as the Hamming window satisfy this property. We will term such a design Filter bank 1 (FB1). With H (f ) and F (f ) having 6 dB cutoff frequencies equal to fs /2K and fs /2M, respectively, and with Nh = Nf = 4K, M = K/2, and K = 512 bins this design achieves a reconstruction error of better than −46 dB for a typical signal of interest such as a broadband BPSK signal. The FB1 design does not satisfy the Parseval criterion though, which will only be satisfied if H (f ) provides −3 dB gain midway between bins. This clearly conflicts with the first requirement. With the illustrative parameter values used above, the power measured by the filter bank is around 80% of the true signal power for a typical broadband signal (see Section 4). A second design class, FB2, uses H (f ) = F (f ) with both filters having 3 dB cutoff frequencies equal to fs /2K. With this design the analysis filter provides the 3 dB attenuation necessary to satisfy the Parseval criterion, while the analysis–synthesis filter combination provides the 6 dB attenuation necessary for perfect reconstruction. The analysis and synthesis filters may be designed by specifying the frequency response of the combined filter and using an inverse Fourier transform to obtain the impulse responses. With an appropriately designed analysis–synthesis pair using Nh = Nf = 4K and the same parameters as used for the previous example, the FB2 design achieves better than −65 dB reconstruction error and the power measured by the analyser is 100% accurate to the first decimal place. Whilst it is desirable to satisfy each of the two criteria discussed above, FRESH filtering applications should satisfy additional criteria if approximate fine frequency shifts are employed. As previously discussed, the approximate fine frequency shift using adjacent bin combining results in noise components being generated that fall outside the bandwidth of the baseband demodulates. This is also the case with the direct approximations to the circular convolution where a truncated transform of the fine shift oscillator is employed. Two potential problems arise from the presence of these noise components: first, the frequency of the noise components could potentially exceed half the decimated

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sampling rate and second, the noise components could destructively contribute to the reconstructed signal. Both of these problems may be minimised by appropriate design of the synthesis filters and the FB2 best satisfies this with appropriate choice of Nf and M. A further complication arises in the bin combining approach when approxiˆ = h(n)(ej φ + ej 2πn/K+j φc ). To achieve this, mating h (n) = h(n)ej 2πn/fs using h(n) we require 1


H (f + ifs /K) = 1

(25)

i=−1

over the bandwidth of h (n). This is satisfied by FB1 but not FB2. As the FB2 design has a −3 dB gain at the midpoint between the analysis filter bank bins, the combination of adjacent bins results in a composite transfer function having gain up to +3 dB at the frequency where the two bins meet. However, this may not be sufficient reason not to use FB2, given that the FB2 synthesis filters provide better noise rejection than FB1.

4. FRESH FILTER PERFORMANCE RESULTS As discussed in Section 2, the performance of a FRESH filter is affected both by the accuracy of the correlation measured between the particular frequency components and also by the amount of corruption introduced into the frequency shifted component. To test the suitability of the various frequency shift techniques to FRESH filtering, we implemented a FRESH filter which utilised N = 2 corrupted inputs plus an ideal reference and designed a test so that with an exact frequency shift, perfect reconstruction was possible (to better than 40 dB output SNR). The filter was implemented using a K = 512 bin DFT filter bank having bin width equal to 156.25 Hz with an 80 kHz sampling rate. The transfer functions were designed according to Eq. (1) with the power spectra estimated using an exponential average of filter bank transform products with a decay coefficient γ = 0.9. A 16 kbaud BPSK signal with square pulse shaping was generated as s(n) and added to a combination of two interferers. The signal plus interference was heterodyned to baseband via inphase and quadrature processing such that at baseband the three interferers are (A) a complex exponential at −4 KHz and (B) complex Gaussian noise source, bandlimited between 3600 and 4400 Hz. The SOI and each of the interferers had unity power. The two inputs to the FRESH filter were the received signal conjugate x ∗ (n) and the received signal shifted by (approximately) one baud rate, denoted x(n) + fb . With the parameters chosen, the baud rate shift may be achieved with a bin rotation equal to 102 bins combined with a 0.4 bin fine shift. This shift is close to the least favorable fine frequency shift of half a bin width. Although this test provided the potential for perfect reconstruction, in the interest of numerical stability we added a small amount, 0.01 in amplitude (−40 dB) of broadband noise, (C) a unity variance

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complex Gaussian noise source, white over the range −fs /2 to fs /2. In addition, we performed a test using a larger amount, 0 dB of (C) to assess the performance of the approaches under a broadband noise condition. The fine shift was approximated by (1) an exact time domain shift, (2) circular convolution, (3) high resolution linear convolution, (4) modified WOLA linear convolution, (5) subband oscillator mixing, (6) adjacent bin combining, and (7) bin rotation only. We also considered a FRESH filter with three signal inputs: x ∗ (n) and x(n) shifted by plus and minus one baud rate, denoted x(n) ± fb . For this N = 3 input FRESH filter, each fine shift was approximated using subband mixing. The reconstruction SNR of the FRESH filter with R = 4 and M = 256 is listed in Table 1. The figures in parentheses are for FB2 and the others correspond to FB1. Figure 5 shows the results for the convolution approaches in greater detail by plotting SNR against the number of oscillator transform points. Figures 5a and 5b show the results for the interference combinations A + 0.01C and A + B + 0.01C, respectively. In each plot the solid lines correspond to the use of filter bank FB2 and the dotted lines, FB1. The upper pairs of lines represent the high resolution convolution and the lower pairs represent the modified WOLA convolution. We will postpone discussion of the performance of the modified WOLA convolution approximations for now and will presently consider the performance of the other techniques. The results clearly show the relative performance of each of the techniques for the cases where the interference is narrowband. The bin rotation is clearly the poorest performer as expected. Subband mixing achieves significantly better results and bin combining improves performance again. The circular convolution achieves the same performance as the exact time domain shift and the high resolution linear convolution approximations achieve progressively better performance as the number of oscillator transform points used in the convolution is increased.

TABLE 1 Output SNR of FRESH Filter Using Approximate Fine Frequency Shift, with Filter Bank FB1 Filters Circular convolution High resolution convolution (15 pts) High resolution convolution (3 pts) High resolution convolution (1 pts) Modified WOLA convolution (15 pts) Modified WOLA convolution (3 pts) Modified WOLA convolution (1 pts) Bin combining Subband mixing Bin rotation only Subband mixing, ± baud shift

A + 0.01C

A + B + 0.01C

A+B+C

42.09 (42.73) 40.78 (41.18) 36.80 (39.36) 35.52 (38.62) 31.30 (32.92) 25.85 (27.32) 21.30 (22.39) 27.54 (31.20) 25.11 (26.21) 19.88 (19.85) 30.91 (35.49)

40.30 (40.70) 38.08 (38.32) 35.03 (37.18) 31.34 (34.13) 25.60 (27.48) 19.48 (21.32) 15.08 (16.22) 23.04 (24.74) 19.77 (20.09) 13.33 (13.31) 26.36 (29.75)

7.22 (7.22) 7.21 (7.22) 7.22 (7.21) 7.22 (7.21) 7.12 (7.16) 6.90 (7.02) 6.43 (6.62) 6.96 (7.05) 6.90 (6.92) 6.29 (6.29) 8.41 (8.46)

Note. The results for filter bank FB2 are given in parentheses.

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(a)

(b) FIG. 5. FRESH filter output SNR versus the number of oscillator transform points used in the convolution.

At low SNR, where there is little uncorrupted redundancy in the filter inputs, Wiener filter theory suggests the best performance is expected from the filter with the greatest number of inputs [4] and this is evidenced by the test with the large broadband interferer, where the three input FRESH filter achieves the best performance. This filter also achieves a performance superior to that of bin combination under the other test conditions. This is expected due to the ability to independently adjust the combination of the bins as opposed

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to simple addition with the bin combining approach. The FRESH filter using plus and minus baud rate shifts also results in less excess bandwidth than does bin combining, particularly for small fine frequency shifts. However, the performance gain comes at the significant cost of extra complexity in the optimal filter. In addition, this approach may only be used to such advantage when both the plus and minus shifted signals are uncorrupted and when the unshifted SOI exhibits correlation with both. An interesting feature of the results is the disparity in the performance between the subband mixing technique and the single point high resolution linear convolution. One might expect similarity between these two techniques and this would be true with R equal to 1. That this is not the case in general for R > 1 and in this particular instance is due to the higher frequency resolution used by the linear convolution approach. With R > 1 a finer bin rotation is a beneficial side effect and the required fine frequency shift is R times less than is required by the other techniques. Of all the improvements in fine shift approximation, this reduction in the fine shift provides the greatest improvement in FRESH filter performance.

4.1. FRESH Filter Performance with Modified WOLA Convolution A surprising result is the performance of the modified WOLA convolution technique. Whilst circular convolution implemented using the modified WOLA architecture achieves the same results as for the exact time domain shift and the high resolution circular convolution, the results worsen considerably as the number of points is reduced in the modified WOLA technique. The modified WOLA requires around nine oscillator transform points in order to achieve performance exceeding that of the bin combining approach. In practice, this may be more points than is computationally feasible to use. The relatively poor performance of the technique may be heuristically explained by considering the K length data segmentation of the WOLA approach. Figure 6a shows a portion of the reconstructed output from a modified WOLA 3 point convolution frequency shift of 19.53 Hz applied to the 16 kHz BPSK SOI with −40 dB wideband noise interferer, with R = 4. Figure 6b shows the similar output using R = 3. The degree of waveform distortion in the R = 4 case is visibly more extreme than with R = 3 and the distortion appears to have the form of an amplitude modulation with a message comprising somewhat discontinuous segments. The segments have shapes which bear distinct resemblance to combinations of the partitioned analysis filter impulse response segments. The segmented impulse responses for the analysis filter of FB1 with K = 512, I = 2, and R = 3 and 4 are shown in Fig. 7. Notice that the central, dominant segment of the R = 3 response has symmetric shape while the central segments of the R = 4 response are asymmetric. This is also true for all other impulse responses with odd and even R, respectively. In the modified WOLA algorithm, the filter segments are multiplied by corresponding signal segments and operated upon by a DFT. The result is then convolved with the truncated DFT of the oscillator. The quality of this approximate circular convolution will depend upon each of the two sequences

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(a)

(b) FIG. 6. BPSK signal reconstruction with WOLA 3 point convolution, K = 512, M = 256, Filter Bank FB1, R = 4 (a), and R = 3 (b).

in the convolution and thus we may expect the DFT of the impulse response segments to provide an indication of the quality expected from the approximate circular convolution. The DFTs of the central segments for the R = 3, 4, 7, 8 analysis filter are shown in Fig. 8 and it can be seen that the response is nearly identical for the two cases of odd R and also for the two cases of even R and that the response is much more dispersed for the even R cases. The concentration of the odd R responses suggests that circular convolution with such sequences

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(a)

(b) FIG. 7. Analysis filter segmentation with K = 512, M = 256, filter bank FB1, R = 4 (a), and R = 3 (b).

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FIG. 8. DFT response of analysis filter central segment with K = 512, M = 256. Shown are the results for R = 3 (lower solid), R = 7 (lower dotted), R = 4 (upper solid), and R = 8 (upper dotted).

using a highly truncated approximation will be more accurate with odd than with even R. In addition, the results suggest nearly identical performance will be achieved for all odd R filters, irrespective of the value of R itself (once the usual filter bank requirements such as aliasing rejection have been met) and likewise for even R filters. In practice, this has been verified with the modified WOLA frequency shift achieving similar reconstruction error for odd cases of R = 3 and R = 7. Similar but considerably poorer results were observed for even cases R = 4 and R = 8. The FRESH filter tests were repeated for the modified WOLA using R = 3 and the results of these tests appear in Fig. 9 along with the SNR of a FRESH filter employing the high resolution convolution, also with R = 3. Figure 9a corresponds to the interference combination A + 0.01C and Fig. 9b corresponds to A + B + 0.01C. The solid lines correspond to the use of filter bank FB2 and the dotted lines FB1. The upper pairs of lines represent the high resolution convolution and the lower pairs represent the modified WOLA convolution. There is a marked improvement in the performance of the FRESH filter with the R = 3 modified WOLA fine shift and the modified WOLA convolution performance now exceeds that of the R = 4 bin combining approach. It should be noted that the modified WOLA approach is the only technique whose performance with R = 3 is superior to that with R = 4. If the filter bank architecture is limited to R = 3 then the relative merit of the modified WOLA is further increased.

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(a)

(b) FIG. 9. FRESH filter output SNR versus the number of oscillator transform points used in the convolution, with R = 3.

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4.2. Frequency Shift Accuracy 4.2.1. 16 kHz frequency shift. In addition to the FRESH filter performance tests, we undertook simulations to evaluate both the mean squared error (mse) and accuracy of correlation measured between the approximately frequency shifted components of x(n) and those of an exact shift. In these tests the uncorrupted BPSK signal was used. For the correlation test, we computed the cross-coherence, averaged across the K bins ρ=

K 1
 K P k=0

P

− α)Xˆ ∗ (m, fk − α) P 2 2 ˆ m=0 |X(m, fk − α)| m=0 |X(m, fk − α)| m=0 X(m, fk

(26)

and for the mse test we computed the signal to noise ratio K P

|(X(m, fk − α))|2 SNR = K P k=0 m=0 . 2 ˆ k=0 m=0 |(X(m, fk − α) − X(m, fk − α))|

(27)

The results for the 16 kHz frequency shift are listed in Table 2. The results were obtained with FB1 and M = 128 as this provides a good pictorial representation, but are to within two decimal points of those obtained with M = 256. As these tests only measure performance of the analysis section, the results using FB1 and FB2 are similar, with the greatest difference being the nonflat response of Hˆ (f ) in the bin combining approach if FB2 is used. The quantitative effect of this is small, however. The results verify expectations with the circular convolution technique providing a perfect fine frequency shift and the coherence of the high resolution linear convolution approximations being very nearly as good. The error power is also good for these techniques. The R = 3, 3-point modified WOLA linear convolution also has good coherence and error performance and as expected, the R = 4 modified WOLA is worse but still performs quite well with 3 oscillator transform points.

TABLE 2 SNR and Coherence between Spectral Components and Approximately Baud Rate Frequency Shifted Components with 16,000 Hz Shift and R = 4 unless Otherwise Indicated Filters Circular convolution High resolution convolution (3 pts) High resolution convolution (1 pt) Modified WOLA convolution (R = 3, 3 pts) Modified WOLA convolution (R = 3, 1 pt) Modified WOLA convolution (3 pts) Modified WOLA convolution (1 pt) Bin combining Subband mixing Bin rotation only

Coherence

SNR (dB)

1.00 0.99 0.98 0.93 0.77 0.89 0.67 0.83 0.73 0.09

>100 17.73 10.55 10.31 4.54 8.28 3.20 1.16 2.75 −3.00

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The SNR for the subband mixing, bin combining, and bin rotation must be carefully interpreted by understanding the sources of the error and the effect of the synthesis filters in reducing this error. To assist this interpretation, the spectra within a particular filter bank bin may be studied. We arbitrarily choose to study bin number 283, which corresponds to a frequency component centred at 4062.5 Hz, within the main lobe of the BPSK spectrum and within the bandwidth of the interferer (B) used in the FRESH filter tests. Figures 10 and 11 depict such spectra for the bin rotation, subband mixing, and bin combining techniques (using a single DFT computed over 388 subband samples, or approximately 50,000 input samples). For comparison, Fig. 11 also shows the corresponding spectrum for a 3-point high resolution convolution using K  = 2048 bins and Fig. 12 depicts the results for the modified WOLA 3-point convolution using R = 3 and R = 4. Figure 10a shows that the error in the subband mixing approach predominantly arises from the frequency components not common to both the exact and the approximate shifts. These comprise approximately 0.4 bin width of high frequency components generated by the approximate shift and a similar amount absent from the negative frequency portion of the subband spectrum. Although not obvious from the plot, there is a high correlation between exact and approximate shift over the subband frequency range from approximately −0.1 to +0.5. Figure 10b relates to the bin combining technique. In this case there is correlation between the exact and approximate shift over the range −0.5 to +0.5 bin. This explains the better correlation for the bin combining technique than for subband mixing alone. In this case, the error arises due to those frequency components of the approximate shift which exceed the bandwidth of the exact shift, namely between −1.1 to −0.5 and +0.5 to +0.9 and this greater error bandwidth explains the lower SNR. The effect of this error must be carefully interpreted as much of the bin combining error will be removed by the action of the synthesis filter, particularly if a narrowband synthesis filter is employed as in FB2. Thus, the SNR, measured over the analysis filter bank bins, does not necessarily imply a higher error in applications such as FRESH filtering. Indeed, the FRESH filter results already discussed demonstrate that the bin combining approach may offer a significant advantage over subband mixing alone. Although it is difficult to see from Fig. 11, there is little correlation between any frequency components of the exact shift and bin rotation approximation. Even though there are no obvious bands where the two spectra do not overlap as with the previous two methods, there is little correlation within the band. Consequently, there is very low mean coherence for this technique, coupled with low SNR. Indeed, the longer the time the two signals are correlated over, the less will be the correlation and in the limit, the coherence tends to zero and the SNR tends to −3 dB, a limit reached in these tests. Figure 11b shows the spectrum for the approximate shift using a high resolution convolution with three points representing the oscillator transform. The spectra of the exact and approximate shifts overlap for the majority of the band and contrary to the coarse bin rotation case, the correlation between

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(a)

(b) FIG. 10. Subband spectrum for exact 16 kHz shift (solid line) and approximate shift using subband mixing (dotted line, a) and bin combining (dotted line, b).

the two spectra is high over this range. Consequently, the mean coherence and SNR are measured to be very high. Note that a principal reason for the good performance here is the high filter bank frequency resolution. The use of more oscillator transform points in the convolution improves the performance further. Figure 12 depicts the spectra for the modified WOLA convolutions and shows that the amount of out-of-bin noise resulting from these approximations is

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(a)

(b) FIG. 11. Subband spectrum for exact 16 kHz shift (solid line) and approximate shift using bin rotation only (dotted line, a) and 3-point high resolution linear convolution (dotted line, b).

relatively high. However, there is high correlation within the bin and the result is the relatively good coherence and SNR measurements. Again, out-of-bin noise will be reduced by the synthesis filters.

4.2.2. Maximum frequency shift. Further tests were conducted to evaluate the frequency shift performance under the most extreme conditions using a fine frequency shift equal to half a bin width. For all techniques other than the high resolution convolution, this corresponds to α = fs /(2K) = 78.125 Hz.

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(a)

(b) FIG. 12. (a) Subband spectrum for 16 kHz shift (solid line) and approximate shift using 3-point modified WOLA linear convolution (dotted line). (b) Corresponding results with R = 3.

The high resolution convolution technique uses α = fs /(2K  ), where K  is the number of bins in the analysis DFT so that K  = RK and α = 19.53 Hz. Additionally, to further aid comparison we also evaluated the other techniques using this 19.53 Hz shift. The results from these tests appear in Table 3. Under the maximum shift conditions, the order of merit is similar to that for the 16 kHz frequency shift, for both performance criteria. However, an interesting anomaly occurs with the subband mixing and bin combining

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TABLE 3 SNR and Coherence between Frequency Shifted Spectral Components Using Exact and Approximate Techniques Filters Circular convolution High resolution convolution (3 pts) High resolution convolution (1 pt) Modified WOLA convolution (R = 3, 3 pts) Modified WOLA convolution (R = 3, 1 pt) Modified WOLA convolution (3 pts) Modified WOLA convolution (1 pt) Bin combining Subband mixing Bin rotation only Modified WOLA convolution (R = 3, 3 pts) Modified WOLA convolution (R = 3, 1 pt) Modified WOLA convolution (3 pts) Modified WOLA convolution (1 pt) Bin combining Subband mixing Bin rotation only

Shift (Hz)

Coherence

SNR (dB)

19.53 19.53 19.53 19.53 19.53 19.53 19.53 19.53 19.53 19.53 78.125 78.125 78.125 78.125 78.125 78.125 78.125

1.00 0.99 0.97 0.97 0.96 0.96 0.94 0.79 0.97 0.11 9.93 0.67 0.89 0.54 0.82 0.62 0.08

>100 16.76 7.66 18.55 14.01 16.39 12.38 0.59 12.13 −3.05 9.69 3.00 7.70 1.82 0.72 0.98 −2.99

Note. R = 4 unless otherwise indicated.

techniques for the 19.53 Hz shift. For this relatively fine shift, the mean coherence measured for the subband mixing technique exceeds that of the bin combining technique. Figure 13 helps show that the reason for this is not a poorer correlation of the bin combining technique, but rather, the reduction in nonoverlapping spectral components of the subband mixing approach improving its performance. This is an example of a circumstance where a better indication of the correlation accuracy would have been to measure the correlation itself, rather than the coherence.

4.3. Discussion of Results The test results clearly show that when it is feasible to use the high resolution convolution approximation, it should be employed. Even when the oscillator transform is approximated by a single frequency point, the results of the FRESH filter studied using this approach are generally bettered only by the exact shift. In an application where a single high performance filter bank processor is available and where the filter bank resolution permits it, this method should be preferred. It does, however, require the filter bank architecture to allow the analyser to operate with higher resolution than the synthesiser. The modified WOLA convolution approximation is more computationally efficient than the high resolution approach, although the performance is not as good when few points are used in the convolution. However, as with the high resolution approach a performance closely approaching that of the exact shift may be obtained if sufficient oscillator transform points are incorporated. This method requires the filter bank to be implemented using the modified WOLA

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(a)

(b) FIG. 13. Subband spectrum for exact 19.53 Hz shift (solid line) and approximate shift using subband mixing (dotted line, a) and bin combining (dotted line, b).

architecture contrasting with the high resolution convolution technique which is independent of the filter bank architecture. The subband mixing approach is particularly simple and computationally efficient and generally provides quite reasonable results. However, for the minimal additional complexity involved, the bin combining modification should generally be used in preference unless the fine shift is particularly small as discussed in Section 4.2.2.

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The coarse bin rotation should never be used on its own unless the required frequency shift is exactly equal to a filter bank bin width. Other than in this instance, the coarse bin rotation performance is very poor. Even the simplest of the more elaborate techniques, the subband mixing, provides an exact frequency shift for a fraction of the spectral components within each bin, which is not possible using only bin rotation. The subband mixing approach, effecting both positive and negative frequency shifts, with both of the resulting shifted signals being processed by a three-input optimal filter may provide performance superior to the bin combining technique in instances where both shifts provide the necessary spectral redundancy and where both shifted signal components are sufficiently free of interference. The extra degree of freedom provided by the additional independent input permits an enhanced performance, exceeding even the exact shift techniques when high power broadband noise is present. The drawback to this approach is the additional complexity required in the optimal filter.

5. SUMMARY AND CONCLUSIONS In this paper we derived the frequency domain operation to achieve an exact shift in frequency of a DFT filter bank input signal for the purpose of optimal FRESH filtering. We have shown this process requires the circular convolution of the signal filter bank transform with the DFT of an appropriate frequency oscillator. We introduced and compared several approximations to the circular convolution shift including two approaches which explicitly approximate the circular convolution by a truncated linear convolution. Other approaches were introduced based on the interpretation of the filter bank analyser as comprising fine frequency shifted bandpass filters. We assessed the performance and developed guidelines for the use of each of the approaches. It was shown that the approach using a high resolution convolution approximation achieves the best shift accuracy. However, the compromise between computational efficiency and shift accuracy is probably best attained by a technique which uses subband mixing and adjacent bin combining.

ACKNOWLEDGMENTS The author thanks Ken Lever, Lang White, and John Tsimbinos for their discussions and various contributions to the work detailed in this paper.

REFERENCES 1. Gardner, W. A., Cyclic Wiener filtering: Theory and method. IEEE Trans. Commun. 41 (1993), 151–163. 2. Crochiere, R. R. and Rabiner, L. R., Multirate Digital Signal Processing. Prentice-Hall, Englewood Cliffs, NJ, 1983. 3. Roberts, S. R., Brown, W. A., and Loomis, H. H., Jr., Computationally efficient algorithms for cyclic spectral analysis. IEEE Signal Process. Mag. April (1991), 38–49.

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4. Ferrara, E. R., Frequency-domain implementations of periodically time-varying filters. IEEE Trans. Acoustics Speech Signal Process. 33 (1985), 883–892. 5. Ferrara, E. R., Frequency-domain adaptive filtering. In Adaptive Filters (Cowan and Grant, Eds.), Chap. 6. Prentice-Hall, Englewood Cliffs, NJ, 1985. 6. Treichler, J. R., Wood, S. L., and Larimore, M. G., Some dynamic properties of transmux-based adaptive filters. In 23rd Asilomar Conference on Signals, Systems and Computers, 1989, pp. 682– 686. 7. Vaidyanathan, P. P., Multirate Systems and Filter Banks. Prentice-Hall, Englewood Cliffs, NJ, 1993.

GARETH PARKER is employed as an engineer with the Defence Science and Technology Organisation of Australia and has recently completed PhD studies at the Institute for Telecommunications Research, University of South Australia. He obtained his honours degree in electrical and electronic engineering from Adelaide University in 1990. His current research interests include signal reconstruction using frequency domain processing techniques.