Widely Linear Filtering and Adaptive Spectrum Estimation

Aims: Students will learn to:
- Implement widely linear adaptive filtering configurations for the estimation of noncircular signals.
- Use widely linear modelling in renewable energy and smart grid applications.
- Use adaptive filters for spectral estimation of non-stationary data.

Complex LMS and Widely Linear Modelling

a) Generate a first-order widely-linear-moving-average process, WLMA(1), driven by circular white Gaussian noise, x(n)

y(n) = x(n) + b₁x(n - 1) + b₂x^*(n - 1) x ~ N (0, 1)

where b₁ = 1.5 + 1j and b₂ = 2.5 - 0.5j. Write a MATLAB function for the ACLMS and implement both the CLMS and ACLMS in the system identification setting to identify the WLMA model in 41. Plot the learning curve, 10log|e(n)|², for the ACLMS and CLMS. Comment on the steady state error of the ACLMS and CLMS.

b) Load the bivariate wind data of the wind speeds in the East-West direction, v_east, and North-South direction, v_north.

Form a complex-valued wind signal

v(n) = v_{east[n] +} jv_north.

for the three wind regimes (low, medium, high). Use the scatter(x,y) function to plot a scatter diagram (the scatter diagram of the real and imaginary parts of a signal is also referred to as a circularity plot) for these three regimes. Comment on the circularity of the complex wind signal for low, medium and high wind speeds. Configure the CLMS and ACLMS filters in a prediction setting to perform a one-step ahead prediction of the complex wind data. Experiment with different filter lengths and comment on which algorithm (CLMS or ACLMS) performs better for the different wind regimes.

Frequency Estimation in Three Phase Power Systems

c) Generate two sets of complex voltages, one balanced and one unbalanced. To generate an unbalanced system, [5] change the magnitude and/or phase of one or more phases. Plot the circularity diagrams of these complex α - β voltages. Comment on the shape of the circularity diagram when the system is balanced vs. unbalanced. How would you use the circularity diagram to identify a fault in the system?

d) Consider the strictly linear and widely linear autoregressive models of order 1, given by

Strictly Linear: v(n + 1) = h*(n)v(n)
Widely Linear: v(n + 1) = h*(n)v(n) + g∗(n)v*(n)

Show that the frequency of the balanced complex α - β voltage in can be derived from the coefficients h(n) in (47) as

f_o(n) = fs/2π arctan{√(ξ{h(n)} - |g(n)|²)/R{h(n)}}

and the frequency of the unbalanced voltage in can be derived from the coefficients h(n) and g(n) in as

f_o(n) = fs/2π arctan{√(ξ²{h(n)})/R{h(n)}}

e) Use the CLMS given and ACLMS algorithms given in (40) to estimate the frequency of the α - β voltages you generated in Part b). For unbalanced system voltages, does the CLMS give the correct frequency estimate? If not, why?

Adaptive AR Model Based Time-Frequency Estimation

a) Generate the frequency modulated (FM) signal y(n) = ej( 2π/fsΦ(n) + η(n) where η(n) is circular complex-valued white noise with zero mean and variance σ₂ = 0.05 and the phase φ(n) = ∫f (n) dn is generated as

                              100,     1 ≤ n ≤ 500

f(n) = dΦ(n)/dn =  100 + (n - 500)/2,     501 ≤ n ≤ 1000

                             100 + ((n - 1000)/25)²,     1001 ≤ n ≤ 1500

Use the aryule function find the AR(1) coefficient for the complete signal of length 1500, then plot the power spectrum of the signal.

b) Implement the CLMS algorithm to estimate the AR coefficient of the signal y(n). At each time instant, compute the frequency spectrum of the signal using the freqz function with the coefficient estimates from the CLMS. Plot the time-frequency spectrum. (Hint: Use the code below.) Comment on the CLMS based spectrum estimate implemented in this part, compared to the stationary AR spectrum in Part a).

A Real Time Spectrum Analyser Using Least Mean Square

a) Show that the least squares solution for the problem in is given by w = (F^HF)^-1 F^Hy and comment on its relationship to the discrete Fourier transform (DFT) formula.

b) Given the least squares interpretation of the DFT, in your own words, explain the Fourier transform in terms of the change of basis and projections.

c) Implement the DFT-CLMS algorithm given in for the frequency modulated signal from Part 4.2 a). Plot the magnitude of the weight vector w(n) at every time instant to create a time-frequency diagram, see Part 4.2 b).

Compare the DFT-CLMS to the adaptive AR-spectrum analyser in Part 4.2. Explain why the spectrum you obtained from the weights of the DFT-CLMS does not resemble to the true power spectrum?

d) Implement the DFT-CLMS for the EEG signal POz used in Part 1.4. To reduce computational burden, choose any segment POz of length 1200, e.g. POz(a:a+1200-1). Explain your observation about the time-frequency spectrum of the EEG signal.