Note: For ELEC4620 assignments, you are required to submit MATLAB code and graphs as well as a well-argued explanation of the steps involved and the reasons for them in your own words. Feel free to demonstrate working MATLAB code to the tutors during the computer laboratory sessions, so that they can mark your work appropriately.
- Derive the Fourier Transform of a rectangular pulse by
a)Differentiating the pulse to form two delta functions
b)Fourier Transforming the derivative
c)Integrating in the frequency domain by dividing by j f 2π .
Repeat this procedure for the triangle function using a double derivative.
Hence or otherwise, show that if a function has discontinuities in the nth derivative, the sidelobes will fall off as 1/f n+1 .(2 marks)
- Multiply the following polynomials in z by using the fft algorithm (MATLAB).
1 + 7𝑧𝑧−1 + 5𝑧𝑧−2 + 13𝑧𝑧−3 + 11𝑧𝑧−4 + 13𝑧𝑧−5
1 − 4𝑧𝑧−1 − 13𝑧𝑧−2 + 11𝑧𝑧−3 − 9𝑧𝑧−4 + 4𝑧𝑧−5 (2 marks)
- Multiply the following base 10 numbers together and then show how the same result can be obtained via the DFT (fft). Explain your method.
8765890 multiplied by 9364267(2 marks)
- Implement the double transform algorithm to Fourier transform two real Npoint sequences using one complex N-point transform. Use the following vectors for your example.
x = [1 2 3 4 5 6 7 8] and y = [1 2 3 4 3 5 3 7]
a) Show that the double transform algorithm gives the same answer as individually transforming the sequences.
b) How should the method be modified if the sequences differ in length?e.g., x = [1 2 3 4 5 6 7 8 9 10] and y = [1 2 3 4 3 5 3 7](2 marks)
5.Consider the polynomial
F(z)= 1 + 2𝑧𝑧−1 + 3𝑧𝑧−2 + 4𝑧𝑧−3 + 3𝑧𝑧−4 + 2𝑧𝑧−5 + 𝑧𝑧−6
a)Write down and plot the positions of all the poles and zeros (MATLAB roots/zplane).
b) Evaluate |F(z)| at 256 uniformly spaced points around the unit circle |z|=1 using the Discrete Fourier Transform and plot the result.(2 marks)
- Consider an 8Hz sine wave sampled at 20Hz. If the samples are plotted and displayed using linear interpolation, you will see that the reconstructed waveform is bumpy and does not resemble the original continuous-time sine wave. Show how sinc interpolation can be used to reconstruct the original continuous-time signal. (Hint: In this question you are effectively being asked to resample the sine wave to a higher sampling rate of, say, 100Hz, by since interpolation – i.e., convolution with a sinc function)(2 marks)
- There is another way to interpolate the sine wave of question 6 in the frequency domain. Fourier transform the sequence and then appropriately zero pad the frequency data before applying the inverse Fourier transform. Show how this can be achieved and compare your answer to the previous result.Explain your method and how you must zero pad the frequency data.(2 marks)
- Determine the number of years in a solar cycle to three decimal places using the annual sunspot data provided on Blackboard. Try to detect other longer cycles in these data using window functions to help resolve weaker signals.Analyse using rectangular, hann, and blackman. windows and comment on the results. Explain which windows are most suitable and why?(6 marks)
Hint: For precision frequency estimation I suggest you try to use the coarse and fine peak search approach of Rife and Boorstyn, Single-Tone Parameter Estimation from Discrete-Time Observations, IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-20, NO. 5, SEPTEMBER 1974. [See Blackboard].(Total 20 marks)