ECE4643/5163 Midterm Practice Exam 2019 Fall
Part 1 Short Answer Questions
1. Given discrete time signals x1[n] and x2[n], and their corresponding system outputs y1[n] and
y2[n], explain how to judge if the discrete time system is time invariant?
2. Given H(z)=
/
, if it is a causal system, what is ROC?
3. Suppose the Z transform of a system is H(z)=
, what is its inverse Z transform?
4. If the sampling rate of a system is 2000Hz, what is the maximum upper band width of the input
signal without aliasing?
5. Suppose an input signal has a upper band limit of 3000Hz, suppose the sampling rate of the
system is 8000 Hz. Design an ideal low pass filter that can restore the sampled signal into the
original continuous time signal without distortion. Plot the ideal filter response and note the upper
band limit of the filter.
6. Find the DTFT of h[n]={1,2,1}. Find its amplitude, phase, and group delay. Is this a linear phase
filter?
7. Find the DTFT H(z)=
/
Find its amplitude, and group delay. Is this a linear phase filter?
8. If a signal X(t) were sampled at 3000Hz, what is its period of its continuous time representation
in the frequency domain?
9. Suppose x[n] has a DTFT of X(), after downsampling by two times, what is the new DTFT?
10. If a discrete time filter has a cutoff frequency of 𝜋/4, and the sampling rate is 4000 Hz, what is
the upper bandwith of the continuous time signal such that the full spectrum of the input signal
can pass the filtering system?
11. Given H(z) =
12. Given H(z) =
13. Given H(z) =
/
/
/
/
Find its zeros and poles.
Find its inverse system. If the inverse system is causal, is it stable?
, find its decomposition of an all-pass component and a minimum-phase
component.
14. Suppose that an LTI has an h[n]={1,3, 3,1}, find its group delay.
15. Suppose a sequence x(t) is first sampled to x[n] with a sampling rate of 4000Hz and the resulted
DTFT is X(w) which is plotted as the following. if it’s up sampled by zero padding by a factor of
two, what’s the resulted frequency spectrum?
16. Suppose the same x(t) in the above problem is sampled with a rate of 8000Hz, what’s the resulted
frequency spectrum?
Part 2: Long questions
Problem 1: A system with feedback is described by: y[n]-1.5y[n-1]+0.56y[n-2]=x[n]. Draw a system
diagram. Find its transfer function H(Z) and find its impulse response h[n]. Find its zero and poles. Is
this system Stable? Causal? Find its all-pass and minimum-phase component if exists. If the input is
x[n]=2 u(n). Find y[n].
Problem 2: Suppose a continuous time aperiodic signal x(t) is one from [-1/1000, 1/1000] second, find its
Fourier transform. Plot its frequency spectrum. If the upper band limit of the anti-alising filter is 2000Hz,
what should be the sampling rate of the filter to prevent alising? Plot the DTFT of its sampled signal after
anti-alising filter, label the periods appropriately. If an ideal low pass filter with an upper band-limit of
1000Hz is applied to the sampled signal, plot the resulted spectrum. Compare the resulted spectrum and
the original spectrum, explain what has caused the difference.