## Physics -Q80Online Services

**Guidelines**

The coursework comprises four assignments, whose individual scores yield 80% of the final mark. The remaining 20%

accounts for presentation and organisation. Students are allowed to discuss the coursework but must code their own

MATLAB scripts, produce their own figures and tables, and provide their own discussion of the coursework assignments.

General directions and notation

The simulations should be coded in MATLAB, a de facto standard in the implementation and validation of signal

processing algorithms.

The report should be clear, well-presented, and should include the answers to the assignments in a chronological

order and with appropriate labelling. Students are encouraged to submit through Blackboard (in PDF format only),

although a hardcopy submission at the undergraduate office will also be accepted.

The report should document the results and the analysis in the assignments, in the form of figures (plots), tables,

and equations, and not by listing MATLAB code as a proof of correct implementation.

The students should use the following notation: boldface lowercase letters (e.g. x) for vectors, lowercase letters

with a (time) argument (x(n)) for scalar realisations of random variables and for elements of a vector, and uppercase

letters (X) for random variables. Column vectors will be assumed unless otherwise stated, that is, x 2 RN1.

In this Coursework, the typewriter font, e.g. mean, is used for MATLAB functions.

**Presentation**

The length limit for the report is 42 pages. This corresponds to ten pages per assignment in addition to one page for front cover and one page for the table of contents, however, there are no page restrictions per assignment but only for the full-report (42 pages).

The final mark also considers the presentation of the report, this includes: legible and correct figures, tables, and captions, appropriate titles, table of contents, and front cover with student information.

The figures and code snippets (only if necessary) included in the report must be carefully chosen, for clarity and to meet the page limit.

Do not insert unnecessary MATLAB code or the statements of the assignment questions in the report.

For figures, (i) decide which type of plot is the most appropriate for each signal (e.g. solid line, non-connected

points, stems), (ii) export figures in a correct format: without grey borders and with legible captions and lines, and (iii) avoid the use of screenshots when providing plots and data, use figures and tables instead.

Avoid terms like good estimate, is (very) close, somewhat similar, etc – use formal language and quantify your

statements (e.g. in dB, seconds, samples, etc).

Note that you should submit two files to Blackboard: the report in PDF format and all the MATLAB code files

compressed in a ZIP/RAR format. Name the MATLAB script files according to the part they correspond to (e.g.

SEASP_Part_X-Y-Z.m).

1 Spectrum Estimation

Aims: Students will learn to

Implement multiple versions of the periodogram and understand their advantages and weaknesses.

Demonstrate and explain the effects of windowing and sampling on spectral leakage and resolution.

Interpret the results when applying non-parametric spectrum estimation techniques on real world data.

In non-parametric spectrum estimation, the signal of interest is passed through a bandpass filter with a narrow bandwidth which is swept through the frequency range. The measure of the spectral contents of the input can then be interpreted as the filter output power divided by the filter bandwidth. One typical non-parametric spectrum estimator is the periodogram.

Background. For a discrete time deterministic sequence fx(n)g, with finite energy

P1

n=1 jx(n)j2 < 1, the Discrete
Time Fourier Transform (DTFT) is defined as
X(!) =
1X
n=1
x(n)e|!n (DTFT): (1)
We often use the symbol X(!) to replace the more cumbersome X(e|!). The corresponding inverse DTFT is given by
x(n) =
1
2
Z
X(!)e|!nd! (inverse DTFT): (2)
This can be verified by substituting (2) into (1). The energy spectral density is then defined as
S(!) = jX(!)j2 (Energy Spectral Density): (3)
A straightforward calculation gives
1
2
Z
S(!)d! =
1
2
Z
1X
n=1
1X
m=1
x(n)x(m)e|!(nm)d! =
1X
n=1
1X
m=1
x(n)x(m)
1
2
Z
e|!(nm)d!
=
1X
n=1
jx(n)j2:
(4)
In the process, we have used the equality
R1
1 e|!(nm)d! = n;m (the Kronecker delta). Equation (4) can be now
restated as
1X
n=1
jx(n)j2 =
1
2
Z
S(!) (Parseval0s theorem): (5)
For random sequences we cannot guarantee finite energy for every realisation (and hence no DTFT). However, a random
signal usually has a finite average power, and can therefore be characterised by average power spectral density (PSD).We assume zero mean data, Efx(n)g = 0, so that the autocovariance function (ACF) of a random signal x(n) is defined as
r(k) = Efx(k)x(k m)g (Autocovariance function ACF): (6)
The Power Spectral Density (PSD) is defined as the DTFT of the ACF in the following way
P(!) =
1X
k=1
r(k)e|!k Denition 1 of Power Spectral Density: (7)
The inverse DTFT of P(!) is given by r(k) = 1
2
R
P(!)e|k!d!, and it is readily verified that 1
2
R
P(!)e|k!d! =
P1
l=1 r(l)
h
1
2
R
e|(kl)!d!
i
= r(k).
Observe that
r(0) =
1
2
Z
P(!)d!: (8)
Since from (6) r(0) = Efjx(n)j2g measures the (average) signal power, the name PSD for P(!) is fully justified, as from
(8) it represents the distribution of the (average) signal power over frequencies. The second definition of PSD is given by
P(!) = lim
N!1
E
8<
:
1
N
NX1
n=0
x(n)e|n!
2
9=
; Denition 2 of Power Spectral Density: (9)
4
1.1 Discrete Fourier Transform Basics
Discrete Fourier Transform (DFT).
As shown in (9), the periodogram can be computed using the DFT coefficients. In this assignment, you will revisit several basic concepts when implementing the DFT in MATLAB and will illustrate the effects of incoherent sampling.
a) Sketch the ideal Fourier (magnitude) spectrum of a 20 Hz sine wave and the theoretical continuous frequency DTFT [5]
(magnitude) spectrum for a windowed sine wave at 20 Hz (with an arbitrary window size).
b) Generate a 20 Hz sine wave of length N = 100, sampled at 1000 Hz, and plot (using stem) the K = 100 point [5]
and K = 1000 point DFT spectra. Comment on whether these plots resemble your sketches in part a).
c) Generate a 24 Hz sine wave of length N = 100, sampled at 1000 Hz, and plot (using stem) the K = 100 point [5]
DFT spectrum and explain why a clear peak at 24 Hz is now not present in the spectrum. (Hint: What is the
frequency resolution for a K = 100 point DFT?) Propose a method to solve this, so called, incoherent sampling
problem.
1.2 Properties of Power Spectral Density (PSD)
Approximation in the definition of PSD.
Show analytically and through simulations that the definition of PSD in (7) is equivalent to that in (9) under a mild [5]
assumption that the covariance sequence r(k) decays rapidly, that is,
lim
N!1
1
N
NX1
k=(N1)
jkjjr(k)j = 0: (10)
Provide a simulation for the case when this equivalence does not hold. Explain the reasons.
Using DTFT to determine PSD from ACF.
Consider an auto covariance function (ACF) given by
r(k) =
Mjkj
M ; jkj M 1
0; otherwise
(11)
Generate r(k) for M = 10 and M = 128 in MATLAB, and form a vector of length L = 256 as:
x =
r(0); r(1); : : : ; r(M 1); 0; : : : ; 0; r(M + 1); : : : ; r(1)
T
(zero padding): (12)
Verify that the MATLAB command xf=fft(x) gives P(!k) for !k = 2k=L (the elements of xf should be nonnegative
and real).
a) Using the zero padding properties of the DTFT, explain whether this particular choice of x is needed for the cases [5]M=10 and M=128. Support your claims by simulation results.
b) The vector xf will often contain a small imaginary component due to round-off errors. Demonstrate, through [5]
simulations, the improved spectral estimate for the case in part a), when the imaginary part of the spectrum is
removed by replacing xf by real(xf).
c) The imaginary part should not be removed unless you are sure it is negligible. Show that the command zf = [5]
real(fft(z)) when applied to
z =
r(M + 1); : : : ; r(1); r(0); r(1); : : : ; r(M 1); 0; : : : ; 0
T (13)
gives erroneous “spectral” values. Explain why this is the case.
d) The use of the fft command gives spectral estimates in the range [0; 2] instead of the more convenient [; ]. [5]
Use the MATLAB command fftshift to exchange the first and second half of the FFT output to correspond to the range [; ]. Similarly, the fftshift can be used to center the ACF around the zero-lag. Experiment with this command for the exercises above. What frequency vector w is needed to ensure that the command plot(w,fftshift(fft(x))) gives the spectral values at correct frequencies? Repeat the exercise for the values of ACF at different time lags for the function stem(n,fftshift(ifft(xf))). Does the result depend on whether the vectors are even or odd in length?
5
1.3 Resolution and Leakage of Periodogram-based Methods
Consider the signal
x(n) = a1 sin
f02n + 1
+ a2 sin
(f0 + =N)2n + 2
+ w(n); w N(0; 2): (14)
where w(n) is real-valued Gaussian noise with zero mean and variance 2, and f0 = 0:2.
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a) Find empirically the 3 dB width of the main lobe of the Bartlett window WB(!) as a function of N, to confirm [10]

that its width is proportional to 1=N. Also, compute the peaks of the sidelobes (in dB) as a function of N. Verify

this through plots of the magnitude of WB(!) for several values of N; use both linear and dB scales in your plots.

b) To eliminate the statistical variation in the periodogram, so that the bias properties can be studied, set 2 = 0. Also, [5]

set a1 = a2 = 1, 1 = 2 = 0 and N = 256. Plot the (zero-padded) periodogram of x(n) for various and

determine the resolution threshold (i.e. the minimum value of for which the two frequency components in (14)

can be resolved). Comment on this value of .

c) Repeat b) for a Hamming-windowed periodogram. [5]

We next analyse the effects of leakage on the periodogram estimate. Leakage is readily visualised when estimating two

sinusoidal components which are well separated in frequency, but have greatly differing amplitudes.

d) Use (14) to generate a sinusoidal sequence for = 4; 2 = 0 and 1 = 2 = 0. Set a1 = 1 and vary a2 (e.g., [5]

a2 2 f1; 0:1; 0:01; 0:001g). Compute the periodogram using a rectangular window and comment on its ability to

identify the second sinusoidal term in the spectral estimate. Does the amplitude threshold for the identification of

the second sinusoidal term change substantially for = 12?

e) Explain your results in d) in connection with the amplitude of the Fourier transform of the Bartlett window at [5]

frequencies corresponding to =N for = 4 and = 12.

f) For practical purposes, we may desire to use a window with a constant sidelobe level which can be chosen by [10]

the user. The Chebyshev window is one such choice (see the MATLAB command chebwin). Repeat d) using

the Chebyshev window to show that the two sinusoidal components in (14) can be resolved using a windowed

periodogram. Compare this result with the Blackman-Tukey method.

1.4 Periodogram-based Methods Applied to Real–World Data Now consider two real–world datasets: a) The sunspot time series1 and b) an electroencephalogram (EEG) experiment.

a) Apply one periodogram-based spectral estimation technique (possibly after some preprocessing) to the sunspot time [10]series. Explain what aspect of the spectral estimate changes when the mean and trend from the data are removed

(use the MATLAB commands mean and detrend). Explain how the perception of the periodicities in the data

changes when the data is transformed by first applying the logarithm to each data sample and then subtracting the

sample mean from this logarithmic data.

The basis for brain computer interface (BCI).

b) The electroencephalogram (EEG) signal was recorded from an electrode located at the posterior/occipital (POz) [10]

region of the head. The subject observed a flashing visual stimulus (flashing at a fixed rate of X Hz, where X is

some integer value in the range [11, . . . , 20]). This induced a response in the EEG, known as the steady state visual

evoked potential (SSVEP), at the same frequency. Spectral analysis is required to determine the value of ‘X’. The

recording is contained in the EEG_Data_Assignment1.mat file2 which contains the following elements:

POz – Vector containing the EEG samples (expressed in Volts) obtained from the POz location on the scalp,

fs – Scalar denoting the sampling frequency (1200 Hz in this case).

Read the readme_Assignment1.txt file for more information.

Apply the standard periodogram approach to the entire recording, as well as the averaged periodogram with different

window lengths (10 s, 5 s, 1 s) to the EEG data. Can you identify the the peaks in the spectrum corresponding

to SSVEP? There should be a peak at the same frequency as the frequency of the flashing stimulus (integer X in

the range [11, . . . , 20]), known as the fundamental frequency response peak, and at some integer multiples of this

value, known as the harmonics of the response. It is important to note that the subject was tired during the recording which induced a strong response within 8-10 Hz (so called alpha-rhythm), this is not the SSVEP. Also note that a power-line interference was induced in the recording apparatus at 50 Hz, and this too is not the SSVEP. To enable a fair comparison across all spectral analysis approaches, you should keep the number of frequency bins the same.

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