ELEN 4810 Final Exam
1. The Z-transform. Consider the following pole-zero diagram:
Please answer the following questions:
(a). Suppose that the system is stable. What is the region of convergence? Is the system causal?
(b). At what frequency or frequencies W is jH(ej! )j maximized?
(c). At what frequency or frequencies W is the group delay grd[H(ej! )] maximized?
(d). Which of the following best characterizes the system?
LOWPASS HIGHPASS ALLPASS BANDPASS BANDSTOP
(e). Suppose we express H(z) as H(z) = Hmin (z)Hap (z), where Hmin is minimum phase, and Hap is all-pass. Please sketch the pole-zero diagrams of Hmin and Happ , labeling poles and zeros.
2. Bilinear Transformation. We design an IIR discrete time filter, by starting from a continuous design Hc (s) with magnitude response Hc (jΩ) as below:
You may assume that this filter was designed to approximate an ideal low-pass filter with cutof frequency Ωc = tan(π/16). We set
and
Please answer the following questions:
(a) Which of the following types of filter is Hc (s)?
BUTTERWORTH CHEBYSCHEV I CHEBYSCHEV II ELLIPTIC
(b) Please sketch jH(ejω )j over the interval [0, π], labelling any cutof frequencies, and indicating which intervals of ω exhibit ripple.
(c) Let β1, . . . , βN be the zeros of Hc (s). Please give an expression for the zeros ζi of H(z) in terms of β1, . . . , βN .
(d) Please give an expression for the impulse response h[n] in terms of the impulse response h1 [n].
(e) Suppose we wish to achieve a sharper transition, without increasing the order of the filter. How can we modify our initial design Hc (s) to accomplish this?
3. FIR Filter Design. Consider two designs of finite impulse response (FIR) bandpass filters. Both designs have a target magnitude response
We plot the gain 20 log1 0jHj and phase \H for each:
Please answer the following questions:
(a). Which filter has the shorter length L?
(b). Both filters are canonical generalized linear phase systems. What type are they, and why?
(c). For the bottom design (H2 ), please use Chebyschev’s alternation theorem to give an upper bound on the length L of the filter.
4. Inverse Systems. Consider an LTI system with impulse response
h[n] = δ[n] — 4δ[n — 2].
Let H(ej! ) denote its frequency response.
Recall that an inverse system for h with impulse response hi [n] satisfies hi * (h * x)[n] = x[n] for any input x with ROC(x) ∩ ROC(h) ∩ ROC(hi ) ≠ 0, and ROC(h) ∩ ROC(hi ) ≠ 0.
(a) Find the impulse response hi [n] of a stable inverse system for h[n]. Is your answer causal?
(b) Find an impulse response g[n] of a system whose magnitude response is the same as jH(ej! )j (i.e., jG(ej! )j = jH(ej! )j for all ω), but which has a causal, stable inverse system.
5. Systems and Spectrograms. A continuous time “chirp” signal
xc (t) = cos(Qt2 ) (10)
is sampled with period T = 10-4s to produce a discrete-time signal x[n] = xc (nT). The signal x[n] is then passed through a stable LTI system with real-valued impulse response h[n] to produce an output signal y[n].
We generate spectrograms of jX[r, k]j and jY [r, k]j with a Hamming window w[n] of length 100, a time step R = 10, and a length-N = 200 DFT. More precisely,
X[r, k] = DFTN {w[n]x[n + rR]} [k]. (11)
Y [r, k] = DFTN {w[n]y[n + rR]} [k]. (12)
The two spectrograms jX[r, k]j and jY [r, k]j are displayed below, for r = 0, . . . 45 and k = 0, . . . 100.
(a) What is the chirp parameter Q? Please make the best estimate you can.
(b) Please sketch the group delay grd(H(ej! ) as a function of ω . Please label the locations and heights of any local maxima or minima. You can use the axis on the following page for your sketch.