Miscellaneous Exercises - Digital Signal Processing: An Introduction with MATLAB and Applications - page 295

x ( n ) y ( n )

z −1

2

z −1

3 − 1

Fig. B.3 Problem Q64

T ½ z ð t Þ¼ z ð t Þ e jp = 2 :
T ae j ð x 1 t þ c Þ

h i þ T be j ð x 2 t þ d Þ h i ¼ ae j ð x 1 t þ c Þ e jp = 2 þ be j ð x 2 t þ d Þ e jp = 2

h i e jp = 2 ¼ T ½ z ð t Þ: Hence, it is linear :

¼ ae j ð x 1 t þ c Þ þ be j ð x 2 t þ d Þ

4. Let y(t) = T[x(t)] = e j(xt + c - p/2) (output of HT). Now

h i ¼ e j x ð t t o Þþ c p = 2

T ½ x ð t t o Þ¼ T e j x ð t t o Þþ c

½ ½ ¼ y ð t t o Þ:

Hence, it is time-invariant.
5. Clearly it is BIBO stable, as it does not affect the magnitude of the signal.

Miscellaneous DSP Exercises—B
Q1: Explain why we cannot use the impulse invariance method to design high
pass digital filters (Hint: due to aliasing).
Q2: Show that the bilinear transformation preserves stability between the analog
prototype filter and the digital filter (Hint: show that the LHS of the s-plane
is transformed inside the unit circle in the z-plane, as shown in the Lecture
Notes).
Q3: Can we use the bilinear transformation to design a digital HPF from an
analog prototype HPF? (Hint: yes, since there is no aliasing).
Q4: Explain

how the analog frequency is transformed using the bilinear

transformation (see Tables).
Q5: Can we use the global average to find the trend of prices in a stock market?
Why? What kind of filter do you suggest for this purpose? Why? (Hint: no,
since the price trend is non-stationary, i.e., time-varying. We use either an
FIR moving average filter, or an alpha filter. We prefer the alpha filter for its
simple structure).
Q6: Explain how we estimate a communication channel transfer function using
an FIR filter.