ECE-5200 Introduction to Digital Signal Processing Autumn 2019

matlab代写 Polyphase Filter Design: There are two different approaches to the design of a polyphase inter- polation filter bank. In the “indirect”

Homework #9 Nov. 8, 2019

HOMEWORK ASSIGNMENT #9 matlab代写

Due Fri. Nov. 15, 2019 (in class)

1.Polyphase Filter Design: There are two different approaches to the design of a polyphase inter- polationfilter bank. matlab代写

In the “indirect” method, we design the “master” filter h[n] for interpolation- by-U and then downsample it to create the polyphase filters, h_p[n] for p = 0, . . . , U − In the “direct” method, we individually design the U  polyphase filters h_p[n] to have DTFTs H_p(e^jω) ≈ e−j ^d ω .

We can compare these two methods by examining the group delays and DTFT magnitude response of the polyphase filters, or by examining the DTFT magnitude response of the corresponding master filter. (Note that we can recover a master filter from the direct design by interleaving the polyphase filters.)matlab代写

In this problem, we will assume interpolation factor U = 15, polyphase filter length L = 11 (i.e., master filter length U L), and an input signal bandlimited to ω₀ = 0.8π radians.

(a)Forthe indirect method, what are the desired passbands and stopbands of H(e^jω)?  Specify your answer for ω ∈ [0, π).matlab代写

(b)Derive an expression for the desired polyphase impulse response, d [n] = F⁻¹(e^−j d−p ω ).Use d = ^U L−1 for compatibility with the master filters designed earlier. Express your answer using sinc(x) ¾  ^sin(πx) , as defined in Matlab (type “help  sinc”).matlab代写

(c)Usingfirls, design a causal, length-U L, generalized-linear-phase weighted-least-squares ap- proximation to the desired h[n]. Then downsample it to create the U polyphase filters h_p[n].

Finally, plot matlab代写

i.The resulting  DTFT  magnitude  response  in  dB  versus  “matlab  normalized  frequency”ω/π ∈ [0, 1].  Use “axis” to zoom into the range between -70 and +30 dB.

ii.Thepolyphase filters’ DTFT magnitude responses in dB versus ω/π ∈ [0, 1].  Use “axis” to zoom into the range between -0.5 and +0.5 dB.

iii.Thepolyphase filters’ group delays (use “grpdelay”).  Use “axis” to zoom into the delay range between 4 and 6.

Use a total of three plots.matlab代写

(d)Designan equiripple approximation to desired h[n] via firpm and give the same plots as (c).

(e)Usethe (Hamming) window design method to directly design each of the length-L polyphase filters h_p[n]. (To do this, simply multiply d_p[n] from (b) by the length-L Hamming ) Next, interleave the polyphase filters h_p[n] into a corresponding master filter h[n]. Finally, give the same plots as (c) and (d).matlab代写

(f)Basedon your plots, which approach do you expect to yield the best interpolator?

2.Polyphase Interpolation: We will now implement a polyphase filterbank in MATLAB using the filters designed in Problem 1.matlab代写

(a)First, generate a length-1050 signal x[n] bandlimited to ω₀= 0.8π radians as follows:

i.Generatea random full-bandwidth signal u[n] using “rng(0);  u  =  randn(1,1000)” and verify that it is full-bandwidth by plotting 20*log10(abs(fft(u,8192))). The command “rng(0)” resets the seed on the random number generator so that results are repeatable.matlab代写

ii.Using firls, design a length-51 lowpass filter g[n] with passband ω ∈ [0, 0.6π) and stopband ω ∈ [0.8π,π).

iii.Filterthe full-bandwidth signal u[n] using conv, and verify that the resulting signal x[n] has bandwidth ω₀ by plotting 20*log10(abs(fft(x,8192))).matlab代写

(b)Generate polyphase-interpolated y[m] using the structure below with U = 15 and the filters designed via the indirect weighted-least-squares approach from Problem 1(c). Demonstrate successfulinterpolation by plotting x[n] for n = [500 : 510] superimposed on the interpolates y[m] (for the suitable range of m).

(c)Repeat(b) for the indirect equiripple design of 1c.

(d)Repeat(b) for the direct Hamming-windowed design of 1e.

(e)Describe the performance of the interpolators. Are they consistent with your observations of the filters from Problem1?matlab代写

Hint: For debugging, recall the equivalence between polyphase and standard interpolation.