Time-dependent Fourier analysis is sometimes implemented as a bank of filters, and even...

Time-dependent Fourier analysis is sometimes implemented as a bank of filters, and even when FFT methods are used, the filter bank interpretation may provide useful insight. This problem examines that interpretation, the basis of which is the fact that when λ is fixed, the time-dependent Fourier transform X[n, λ), defined by Eq. (10.18), is simply a sequence that can be viewed as the result of a combination of filtering and modulation operations.

(a) Show that X[n, λ) is the output of the system of Figure P10.46-1 if the impulse response of the LTI system is h₀[n] = w[−n]. Show also that if λ is fixed, the overall system in Figure P10.46-1 behaves as an LTI system, and determine the impulse response and frequency response of the equivalent LTI system.

(b) Assuming λ fixed in Figure P10.46-1, show that, for typical window sequences and for fixed λ, the sequence has a low-pass DTFT. Show also that, for typical window sequences, the frequency response of the overall system in Figure P10.46 is a band-pass filter centered at ω = λ.

(c) Figure P10.46-2 shows a bank of N bandpass filter channels, where each channel is implemented as in Figure P10.46-1. The center frequencies of the channels are λ_k = 2πk/N, and h₀[n] = w[−n] is the impulse response of a lowpass filter. Show that the individual outputs y_k [n] are samples (in the λ-dimension) of the time-dependent Fourier transform. Show also that the overall output is y[n] = Nw[0]x[n]; i.e., show that the system of Figure P10.46-2 reconstructs the input exactly (within a scale factor) from the sampled time-dependent Fourier transform.

The system of Figure P10.46-2 converts the single input sequence x[n] into N sequences, thereby increasing the total number of samples per second by the factor N. As shown in part (b), for typical window sequences, the channel signals have lowpass Fourier transforms. Thus, it should be possible to reduce the sampling rate of these signals, as shown in Figure P10.46-3. In particular, if the sampling rate is reduced by a factor R = N, the total number of samples per second is the same as for x[n]. In this case, the filter bank is said to be critically sampled. (See Crochiere and Rabiner, 1983.) Reconstruction of the original signal from the decimated channel signals requires interpolation as shown. Clearly, it is of interest to determine how well the original input x[n] can be reconstructed by the system.

(d) For the system of Figure P10.46-3, show that the regular DTFT of the output is given by the relation

where λ_k = 2πk/N. This expression clearly shows the aliasing resulting from the decimation of the channel signals From the expression for Y(e^jω), determine a relation or set of relations that must be satisfied jointly by H₀(e^jω) and G₀(e^jω) such that the aliasing cancels and y[n] = x[n].

(e) Assume that R = N and the frequency response of the lowpass filter is an ideal lowpass filter with frequency response

For this frequency response H₀(e^jω), determine whether it is possible to find a frequency response of the interpolation filter G₀(e^jω) such that the condition derived in part (d) is satisfied. If so, determine G₀(e^jω).

(f) Optional: Explore the possibility of exact reconstruction when the frequency response of the lowpass filterH₀(e^jω) (the Fourier transform of w[−n]) is nonideal and nonzero in the interval |ω| < 2π/N.

(g) Show that the output of the system of Figure P10.46-3 is

From this expression, determine a relation or set of relations that must be satisfied jointly by h₀[n] and g₀[n] such that y[n] = x[n].

(h) Assume that R = N and the impulse response of the lowpass filter is

For this impulse response h₀[n], determine whether it is possible to find an impulse response of the interpolation filter g0[n] such that the condition derived in part (g) is satisfied. If so, determine g₀[n]. (i) Optional: Explore the possibility of exact reconstruction when the impulse response of the lowpass filter h₀[n] = w[−n] is a tapered window with length greater than N.