ECE438 Week8 Quiz - Rhea

Under construction -Jaemin

Quiz Questions Pool for Week 8

Q1. Find the impulse response of the following LTI systems and draw their block diagram.

(assume that the impulse response is causal and zero when $$ n<0 $$ )

${\color{White}ab}\text{a)}{\color{White}abc}y[n] = 0.6 y[n-1] + 0.4 x[n]$

${\color{White}ab}\text{b)}{\color{White}abc}y[n] = y[n-1] + 0.25(x[n] - x[n-3])$

Solution.

Q2. Suppose that the LTI filter $$ h_1 $$ satifies the following difference equation between input $$ x[n] $$ and output $$ y[n] $$ .

${\color{White}ab} y[n] = h_1[n]\;\ast\;x[n] = \frac{1}{4} y[n-1] + x[n]$

( $\ast$ implies the convolution)

Then, find the inverse LTI filter $$ h_2 $$ of $$ h_1 $$ , which satisfies the following relationship for any discrete-time signal $$ x[n] $$ ,

(assume that the impulse responses are causal and zero when $$ n<0 $$ )

${\color{White}ab} x[n] = h_2[n]\;\ast\;h_1[n]\;\ast\;x[n]$

Solution.

$\text{Q3.}$

The following figure shows the flow diagram that results for an N=8 FFT algorithm. The bolded line indicates a path from input sample x[7] to DFT sample X[2].

a) What is the gain of the path?

b) How many paths exist beginning at x[7] and ending up at X[2]? Does the result apply to a general condition? i.e. How many paths are there between every input sample and output sample?

c) Consider DFT sample X[2]. Following paths displayed in the flow diagram. Prove that every input sample contributes the proper amount to the output DFT sample.

i.e. $X[2]=\sum_{n=0}^{N-1} x[n]e^{-j(2\pi /N)2n}$