After a discrete-time signal is lowpass filtered, it is often downsampled or decimated,...

After a discrete-time signal is lowpass filtered, it is often downsampled or decimated, as depicted in Figure P7.56-1. Linear-phase FIR filters are frequently desirable in such applications, but if the lowpass filter in the figure has a narrow transition band, an FIR system will have a long impulse response and thus will require a large number of multiplications and additions per output sample.

In this problem, we will study the merits of a multistage implementation of the system in Figure P7.56-1. Such implementations are particularly useful when ω_s is small and the decimation factor M is large. A general multistage implementation is depicted in Figure P7.56-2. The strategy is to use a wider transition band in the lowpass filters of the earlier stages, thereby reducing the length of the required filter impulse responses in those stages. As decimation occurs, the number of signal samples is reduced, and we can progressively decrease the widths of the transition bands of the filters that operate on the decimated signal. In this manner, the overall number of computations required to implement the decimator may be reduced.

(a) If no aliasing is to occur as a result of the decimation in Figure P7.56-1, what is the maximum allowable decimation factor M in terms of ω_s?

(b) Let M = 100, ω_s = π/100, and ω_p = 0.9π/100 in the system of Figure P7.56-2. If x[n] = δ[n], sketch V (e^jω) and Y (e^jω).

Now consider a two-stage implementation of the decimator for M = 100, as depicted in Figure P7.56-3, where M₁ = 50, M₂ = 2, ω_p1 = 0.9π/100, ω_p2 = 0.9π/2, and ω_s2 = π/2. We must choose ω_s1 or, equivalently, the transition band of LPF₁, (ω_s1−ω_p1), such that the two-stage implementation yields the same equivalent passband and stopband frequencies as the single-stage decimator. (We are not concerned about the detailed shape of the frequency response in the transition band, except that both systems should have a monotonically decreasing response in the transition band.)

(c) For an arbitrary ω_s1 and the input x[n] = δ[n], sketch V₁(e^jω), W₁(e^jω), V₂(e^jω), and Y (e^jω) for the two-stage decimator of Figure P7.56-3.

(d) Find the largest value of ω_s1 such that the two-stage decimator yields the same equivalent passband and stopband cutoff frequencies as the single-stage system in part (b).

In addition to possessing a nonzero transition bandwidth, the lowpass filters must differ from the ideal by passband and stopband approximation errors of δ_p and δ_s , respectively. Assume that linear-phase equiripple FIR approximations are used. It follows from Eq. (7.117) that, for optimum lowpass filters,

Where N is the length of the impulse response and ω = ω_s−ω_p is the transition band of the lowpass filter. Equation P7.56-1 provides the basis for comparing the two implementations of the decimator. Equation (7.76) could be used in place of Eq. (P7.56-1) to estimate the impulse-response length if the filters are designed by the Kaiser window method.

(e) Assume that δ_p = 0.01 and δ_s = 0.001 for the lowpass filter in the single-stage implementation. Compute the length N of the impulse response of the lowpass filter, and determine the number of multiplications required to compute each sample of the output. Take advantage of the symmetry of the impulse response of the linear-phase FIR system. (Note that in this decimation application, only every M^th sample of the output need be computed; i.e., the compressor commutes with the multiplications of the FIR system.)

(f) Using the value of ω_s1 found in part (d), compute the impulse response lengths N₁ and N₂ of LPF₁ and LPF₂, respectively, in the two-stage decimator of Figure P7.56-3. Determine the total number of multiplications required to compute each sample of the output in the two-stage decimator.

(g) If the approximation error specifications δ_p = 0.01 and δ_s = 0.001 are used for both filters in the two-stage decimator, the overall passband ripple may be greater than 0.01, since the passband ripples of the two stages can reinforce each other; e.g., (1 + δ_p)(1 + δ_p) > (1 + δ_p). To compensate for this, the filters in the two-stage implementation can each be designed to have only one-half the passband ripple of the single-stage implementation. Therefore, assume that δ_p = 0.005 and δ_s = 0.001 for each filter in the two-stage decimator. Calculate the impulse response lengths N₁ and N₂ of LPF₁ and LPF₂, respectively, and determine the total number of multiplications required to compute each sample of the output.

(h) Should we also reduce the specification on the stopband approximation error for the filters in the two-stage decimator?

(i) Optional. The combination of M₁ = 50 and M₂ = 2 may not yield the smallest total number of multiplications per output sample. Other integer choices forM₁ andM₂ are possible such that M₁M₂ = 100. Determine the values of M₁ and M₂ that minimize the number of multiplications per output sample.