This is an assignment of the course Advanced Signal Processing (AUTh-8th semester)
At the first part of the assignment (questions 1-6) the aim is to construct a MA-q system (for q=5) with known input and output, in order to check the validity of Giannakis' formula.
In order to do so, the following steps were conducted:
- Initially, a real discrete signal x[k] is constructed, with k=1,2, ...,N=2048, which is derived as the output of a MA-q process with coefficients of [1, 0.93, 0.85, 0.72, 0.59, -0.1], driven by non-Gaussian noise v[k], which is derived from an exponential distribution with mean value of 1 (file X.m).
- The non-Gaussian character of input v[k] is justified by calculating its skewness (question 1 - file skew.m).
- The
$3^{rd}$ order cumulants of x[k] are estimated and plotted using the indirect method, with K=32, M=64 and L3=20 (question 2 - file cum3.m) - The
$3^{rd}$ order cumulants were used to estimate the impulse response of the MA system, using the Giannakis' formula, yet considering sub-estimation of the order q (qsub = q-2), sup-estimation of the order q (q_sup = q+3) (questions 3,4 - file h_est.m) - Finally, the MA-q system output output is estimated for each impulse response (file x_est.m), along with the Normal Root Mean Square Error (questions 5,6 - file NRMSE_fun.m)
The demonstration of the process takes place in the demo_x.m file
In the following 3 images, the original signal x[k] is depicted, along with the estimated, the sub-estimated and the sup-estimated respectively.
| X estimated | X sub_estimated | X sup_estimated |
|---|---|---|
The NRMSE for each case can be seen in the following table:
| h_est | h_sub | h_sup |
|---|---|---|
| 0.1531 | 0.1709 | 0.2609 |
At the second part of the assignment a source of white Gaussian noise is added at the output of the system, producing a variation in the SNR of [30:-5:-5]dB. Then, the same process is followed as the first part, but instead of x[k], the noise contaminated output is used, for each level of SNR (question 7 - demo_y.m file).
In the following pictures the original signal y[k] is depicted along with the estimated y_est[k] for SNR [30:-5:15]dB and [10:-5:-5]dB respectively.
| SNR [30:-5:15]dB | SNR [10:-5:-5]dB |
|---|---|
The signal estimation seems to improve for higher SNR levels.
At the last part, instead of using one realization of the input and output data of the MA-q system, the whole process is repeated 50 times using the mean values. Thus, more valid conclusions can be drawn regarding Giannakis' formula reliability (question 8 - meanValues.m file).
The following image depicts the boxplots of the mean NRMSE for the real order q, the sub-estimation of the order q-2 and the sup-estimation of the order q+3. Obviously, by sub-estimating the order, the width of the boxplot is much narrower and close to 0, compared to the rest plots.
Summarizing, Giannakis' formula can be applied in practice, making it possible to
calculate the impulse response of a system with only the information of
Note: files cumest.m and cum3est.m are derived from the HOSA TOOLBOX.