Matlab代写 - ELEC3104: Digital Signal Processing

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 GRAPHICAL WORK. 1 Question 1. (20 marks) A continuous time signal, 𝑥ሺ𝑡ሻ = cosሺ800𝜋𝑡ሻ + 2 cosሺ1200𝜋𝑡ሻ, is to be sampled to obtain a discrete time signal, 𝑥ሾ𝑛ሿ. A. What sampling rates could you choose to avoid aliasing? [2 Marks] B. If the sampling rate was chosen to be 2000Hz, give an expression for 𝑥ሾ𝑛ሿ. [3 Marks] C. Find 𝑥ොሺ𝜃ሻ, the DTFT of 𝑥ሾ𝑛ሿ, and justify your answer with appropriate equations. [10 Marks] D. If 𝑥ሺ𝑡ሻ was instead sampled at 1000Hz, determine 𝑥ሾ𝑛ሿ and sketch |𝑥ොሺ𝜃ሻ|. [5 Marks] Question 2. (20 marks) If the impulse response, ℎሾ𝑛ሿ, of a linear time-invariant system is given by: ℎሾ𝑛ሿ = 𝛼௡𝑢ሾ𝑛ሿ + 𝛽௡𝑢ሾ1 − 𝑛ሿ where, 𝑢ሾ𝑛ሿ denotes the unit step function and 𝛼 and 𝛽 are positive real numbers A. Determine the transfer function, 𝐻ሺ𝑧ሻ and the corresponding ROC (region of convergence) by computing the z-transform of ℎሾ𝑛ሿ [10 Marks] B. For what values of 𝛼 and 𝛽 will the system be stable? [5 Marks] C. What are the constraints on 𝛼 and 𝛽 that will make this system causal? [5 Marks] 2 Question 3. (25 marks) One approach to designing a notch filter to filter out a specific frequency (notch frequency - 𝑓ே) is to place a pair of zeros on the unit circle at the angles corresponding to the notch frequency, 𝜃ே = ±2𝜋𝑓ே/𝐹௦ 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: 𝑟 = 1 − ൬ Δ𝑓 𝐹௦ ൰ 𝜋 where, Δ𝑓 is the desired 3dB bandwidth and 𝐹௦ 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 [4 Marks] B. Give the transfer function of this filter, such that the DC gain is one. [7 Marks] C. Sketch the Direct Form II implementation of this filter [6 Marks] D. If the input to this filter is the discrete-time signal obtained by sampling 𝑥ሺ𝑡ሻ = cosሺ280𝜋𝑡ሻ 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. [8 Marks] 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 characteristics. A. Write down an expression for the desired frequency response, ℎ෠ௗሺ𝜃ሻ and sketch the desired magnitude response, หℎ෠ௗሺ𝜃ሻห. [4 Marks] B. From 𝑥ොௗሺ𝜃ሻ, derive an expression for the desired impulse response, ℎௗሾ𝑛ሿ [8 Marks] 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. [5 Marks] D. Sketch the Direct Form I implementation of this filter [3 Marks] 3 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. [8 Marks] B. Determine the transfer function of this system [3 Marks] C. Sketch the Direct Form I implementation of this system [4 Marks] 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. [5 Marks] End of paper