3551 Trousdale Rkwy, University Park, Los Angeles, CA
1. Time allowed: 2 hours.
2. The paper contains 5 questions.
3. The questions are worth 105 marks in total, your final mark will be capped at 100.
4. The questions are NOT of equal value. Part marks are as indicated.
5. This question booklet contains 3 pages apart from this instructions page.
ANSWERS MUST BE WRITTEN IN INK. EXCEPT WHERE THEY ARE EXPRESSLY
REQUIRED, PENCILS MAY ONLY BE USED FOR DRAWING, SKETCHING OR
Question 1. (20 marks)
A continuous time signal, ሺሻ = cosሺ800ሻ+ 2 cosሺ1200ሻ, is to be sampled to obtain
a discrete time signal, ሾሿ.
A. What sampling rates could you choose to avoid aliasing?
B. If the sampling rate was chosen to be 2000Hz, give an expression for ሾሿ.
C. Find ොሺሻ, the DTFT of ሾሿ, and justify your answer with appropriate equations.
D. If ሺሻ was instead sampled at 1000Hz, determine ሾሿ and sketch |ොሺሻ|.
Question 2. (20 marks)
If the impulse response, ℎሾሿ, of a linear time-invariant system is given by:
ℎሾሿ = ሾሿ+ ሾ1− ሿ
where, ሾሿ denotes the unit step function and and are positive real numbers
A. Determine the transfer function, ሺሻ and the corresponding ROC (region of
convergence) by computing the z-transform of ℎሾሿ
B. For what values of and will the system be stable?
C. What are the constraints on and that will make this system causal?
Question 3. (25 marks)
One approach to designing a notch filter to filter out a specific frequency (notch frequency -
ே) is to place a pair of zeros on the unit circle at the angles corresponding to the notch
frequency, ே = ±2ே/௦ and a pair of poles at the same angles but slightly inside the unit
circle. The radius at which the poles are placed is given as:
= 1 − ൬Δ௦ ൰
where, Δ is the desired 3dB bandwidth and ௦ denotes the sampling rate.
Using this method, design a notch filter with a 3dB bandwidth of 10Hz to remove the 125Hz
component from a signal sampled at 1kHz.
A. Sketch the pole-zero plot of this notch filter
B. Give the transfer function of this filter, such that the DC gain is one.
C. Sketch the Direct Form II implementation of this filter
D. If the input to this filter is the discrete-time signal obtained by sampling ሺሻ =cosሺ280ሻ at a sampling rate of 1 kHz, how much would it be attenuated by as it
goes through the filter? Provide your answer in dB.
Question 4. (20 marks)
A causal discrete time 6th order FIR high-pass filter operating at a sampling rate of 10 kHz
and with a cut-off frequency of 4 kHz is required. The filter must also have linear phase
A. Write down an expression for the desired frequency response, ℎௗሺሻ and sketch the
desired magnitude response, หℎௗሺሻห.
B. From ොௗሺሻ, derive an expression for the desired impulse response, ℎௗሾሿ
C. Obtain the coefficients of a 6th order FIR filter by multiplying the desired impulse
response by a suitable rectangular window and write down the transfer function.
D. Sketch the Direct Form I implementation of this filter
Question 5. (20 marks)
A MATLAB function is provided to you on Moodle in the form of an encrypted p-file (i.e.,
you cannot open it and look at the code) named ‘model_q’. It is a discrete-time system that
you can run by calling it as a MATLAB function, i.e., ‘y = model_q(x)’ will give you the
output, y, of the system ‘model_q’ for the input signal x.
A. Sketch the pole-zero plot corresponding to this system.
B. Determine the transfer function of this system
C. Sketch the Direct Form I implementation of this system
D. Is this system a good notch filter? If so, please explain your reasons and describe
what properties it possess that makes it a good notch filter. If not, briefly explain
what changes you could make to the system, in terms of introducing new poles or
zeros, to make it a better notch filter.
End of paper