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**Unformatted text preview: **EE 4541 Digital Signal Processing Fall 2007 Problem Set Solutions 1. In the system shown below, ( ) H e jω ⎧ − jω
⎪4e ,
=⎨
⎪ 0,
⎩ |ω | < π
8 π
8 <| ω | < π a) (10 points) Sketch the spectra of v[n] , w[n] , and y[n] given the spectrum of x[n] shown in the figure. b) Assuming x[n] = xc (nT ) , can you express y[n] in terms of xc (nT ′) , where T ′ is appropriately chosen based on the upsampling and downsampling factors? If yes, give this relationship. If not, justify your answer. x[n] ↑ 4 h[n] v[n] ↓ 6 w[n] 1 ( ) X e jω −π − π π 2 2 π y[n] EE 4541 Digital Signal Processing Fall 2007 Problem Set Solutions EE 4541 Digital Signal Processing Fall 2007 Problem Set Solutions 2. Consider the system shown in the figure below. It represents an oversampling A/D system with noise shaping. The noise source, , is an additive model of the quantization source. Read Section 6.6 in P & M for another example of noise‐shaping filter in oversampling Σ‐Δ A/D. a) Express the output Y(z) in the form: Y ( z ) = H x ( z ) X ( z ) + H v ( z )V ( z ) . Find Hx(z) and Hv(z) in terms of H(z). Here we define the transfer functions as follows: Y ( z)
Y ( z)
H v ( z) =
. H x ( z ) =
X ( z ) V ( z )=0
V ( z ) X ( z )=0
b) Evaluate these transfer functions for the special case H ( z ) =
y[n] = y x [n] + yv [n] 1
. Show that 1 − z −1 = x[n] + yv [n]
where y x [n] & y v [n] are the outputs due to x[ n] & v[ n] , respectively. What is y v [n] in terms of v[n] ? c) (Graduate Students) Assume that v[n] is zero‐mean white stationary noise with variance σ v2
. Show that the power spectral density of the noise component of y[n] is given by: S yv yv e jω = 16σ v2 sin 4 (ω / 2) ( ) d) Assume that the power spectral density of x[n ] is as given in the figure below. Sketch the spectral densities of the signal and noise components of y[n] = y x [n] + yv [n] and yd [n] = ydx [n] + ydv [n] . Briefly comment on the significance of your results. [Undergraduate Students: solve Part d) assuming the statements given in Parts b) and c) are correct even if you are unable to prove these statements.] v[n] x[n] H(z) Σ Σ Σ Ideal LPF
3. π
ωc =
M y[n] z‐1 y[n] H(z) ( ) S xx e jω y f [n] ↓M y d [ n] = y f [ Mn] −π − π π M M
1 π EE 4541 Digital Signal Processing Fall 2007 Problem Set Solutions EE 4541 Digital Signal Processing Fall 2007 Problem Set Solutions EE 4541 Digital Signal Processing Fall 2007 Problem Set Solutions 3. Consider the following sampling scenarios for audio applications: The first scenario is the conventional Nyquist sampling one while the second represents the "4x ( ) oversampling" scenario. In the latter, H e jω is an ideal lowpass digital filter with cutoff ↓4 frequency at ω c = π / 4 and denotes a "downsampling" operation (In the figure below, x 2 [n] = y o [4n] ). Answer the following questions about the two sampling systems: a) What is the cutoff frequency for each of the two antialiasing filters (assumed ideal)? b) For a sinusoidal input signal of frequency 5.5 kHz, what is the relative frequency of all the resulting digital sequences shown for both sampling systems? c) Repeat Part b) for an input frequency of 44 kHz. d) Comment on the role of the digital lowpass filter in the 4x system. e) (Bonus 25 points) Based on your answers above, discuss how the specifications of the antialiasing filter can be simplified for the 4x system. Consider only the edge frequencies for the passband and stopband in each case. x a (t ) H a1 ( jΩ) x1[n] A/D (1/T1)= 44 kHz x a (t ) H a 2 ( jΩ ) xo [n] A/D (1/T2)= 4X44 ( ) H e jω y o [n] ↓4 x2[ n] EE 4541 Digital Signal Processing Fall 2007 Problem Set Solutions EE 4541 Digital Signal Processing Fall 2007 Problem Set Solutions ...

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