ECE-465
Midterm
May 6, 2013
1. Consider the following signal
x(t) = Sin(10000t)+Sin(20000t)+Sin(60000t)
The signal is pre-filtered by an anti-aliasing filter H(f), and then sampled
at the sampling rate of 40,000 Hz. The samples are then reconstructed by
an ideal reconstruction filter to reproduce the analog signal. Find the
output signal for the following cases:
(a) H(f) = 1, i.e., no pre-filtering.
(b) H(f) is an ideal low pass filter with cutoff frequency at 20,000 Hz.
(c) H(f) is a filter with flat response up to 20,000 Hz, and attenuate at the
rate of 60dB per Octave beyond 20,000 Hz.
2. An analog input signal to a DSP system has a spectrum
|Xin(f)| = 1/sqrt[1+(f/1000)8)]
where f is in Hertz. The frequency range of interest is from DC to 2 kHz.
The signal is to be sampled at the rate of fs Hertz.
(a) If the sampling rate is 10 kHz, what is the smallest signal to aliasing
noise ratio (in terms of dB) in the frequency range of interest?
(b) If one desires to achieve at least 80 dB signal to noise ratio, what is the
minimum sampling frequency should be?
3. In a DSP system, an analog signal is reconstructed by a D/A converter
with an exponential impulse, i.e., h(t) = exp(-t)u(t) followed by an ideal
anti-image filter.
(a) Show that the Fourier transform of h(t) is H(f) = 1/(1+j2f).
(b) If a signal x(t) = Cos(4000t) is sampled at 20,000 Hz and
reconstructed at the same clock rate, what is the Fourier transform of the
output signal.
(c) Find the output signal in time domain.
4. An LTI system has the following impulse response
(a) h[n] = (-1)nu[n]. Is the system stable? Why?
(b) Repeat (a) with h[n] = (0.9)n Cos(n /5)u[n]. Prove your result.
5. An LTI system has a transfer function:
H(z) = (3+0.1z-1)/(1-0.5 z-1-0.66 z-2 ).
(a) Find the causal impulse response h[n].
(b) Find the stable impulse response h[n].