Digital Signal Processing PW1 Signals, Correlation functions and Spectra

Transcription

1 Digital Signal Processing PW1 Signals, Correlation functions and Spectra Nathalie Thomas Master SATCOM Introduction The objectives of this rst practical work are the following ones : 1. to be able to generate a simple digital signal (cosine or sum of cosines) and plot it with a time scale in seconds.. to be able to generate a noisy digital signal using a white Gaussian noise and for a given signal to noise ratio. 3. to be able to estimate a digital signal autocorrelation function and plot it with a time scale in seconds. 4. to be able to estimate a digital signal power spectral density (PSD) and plot it with a frequency scale in Hz and with a frequency scale in normalized frequencies. 5. to be able to use the autocorrelation and PSD estimations in order to retrieve the signal parameters for some simple cases. 1

2 Guidelines : 1. All the plots must have labels on the axis (use xlabel and ylabel with matlab) and a title (use title with matlab).. If several curves are plotted on the same gure you must put a legend (use legend with matlab). 3. You must put enough and clear comments on your codes. A new user must understand. Documents available at http ://thomas.perso.enseeiht.fr/teaching.html : 1. Some slides, with code and plots examples.. A course handout, with the denitions of used estimators. 3. An exercise handout, with the DFT computation for a complex exponential function (reminder : cos x = ejx +e jx ). 4. Some elements on Matlab programming. Digital signal generation and plotting 1. Generate 10 periods of a cosine (function cos on matlab), with an amplitude 1 (V), a frequency f 0 = 100 Hz and sampled at F e = Hz.. Plot the generated cosine with a time scale in seconds. You must be able to retrieve the good value for the cosine frequency from the plot. 3 Digital noise generation and plotting 1. Generate a white Gaussian noise (WGN) of the same length as the previously generated cosine, with a power equal to 1, using the Matlab randn function. σ randn(1,n) allows to generate N samples of Gaussien noise with a power σ.. Plot the generated noise with a time scale in seconds.

3 4 Digital noisy signal generation and plotting Add some noise (WGN) to the generated cosine in order to obtain a noisy cosine with a signal to noise ratio equal to 10 db. Plot the generated noisy cosine with a time scale in seconds. We give : ( ) Ps SNR db = 10 log 10 P n where SNR db is the signal to noise ratio given in decibels, P s represents the signal power and can be computed on Matlab by using, for example, the following code : P s = mean (abs(x).ˆ) if x is the signal samples vector (function mean computes the vector average, function abs compute the vector modulus and.ˆ is used to put to the square the vector elements). P n must be determined from P s to meet the wanted signal to noise ratio. 5 Estimation and plotting of a digital autocorrelation function We will used a biased estimator of the autocorrelation function (see course handout). If we remark that this estimator can be written like : R x (k) = 1 N x(k) x ( k) for a given signal x, then it is possible to use the matlab function allowing to compute a convolution (conv.m), a conjugated (conj.m) and a vector reversal (iplr.m) to implement it. It is also possible to code the autocorrelation expression or to use the Matlab function xcorr.m (giving a biased or unbiased estimation depending on the used parameter). 1. Estimate and plot, with a time scale in seconds, the biased estimation of the previously generated noisy cosine autocorrelation function. (a) Why do we call this estimator "biased" estimator? (b) Is it a problem to have a biased estimation?. Modify the noise power in order to generate a noisy cosine with a signal to noise ratio SNR db = 10 db. Plot the generated noisy cosine with a time scale in seconds, as well as its biased autocorrelation function. 3

4 (a) Can you see an interest to estimate the autocorrelation function in this case? (b) Retrieve, from the estimation of the autocorrelation function, the frequency and power of the cosine and the power of the added noise. 6 Estimation and plotting of a digital power spectral density 1. Estimate the power spectral density of the previously generated noisy cosine using a periodogram and a biased correlogram (see the course handout).. Plot both estimations, with a frequency scale in Hz, on the same gure. You can plot them with a log scale using semilogy function instead of plot. What do you notice? Explain the obtained result. 3. Plot both estimations with a frequency scale in normalized frequencies. 4. Retrieve, from the PSD estimations, the frequency and the power of the cosine and the noise power. 7 Annex 7.1 Some theoretical correlations Cosine function with a frequency f 0 and a random phase φ uniformly distributed on [0, π] : White noise : x(t) = A cos(πf 0 t + φ) R x (τ) = A cos(πf 0τ) (1) where δ(τ) represents a Dirac : with δ(τ)dτ = 1. R δ(τ) = R x (τ) = N 0 δ(τ) () { τ = 0 0 τ 0 4

5 7. Noisy cosine periodogram Let's consider N samples of a cosine of frequency f 0 and amplitude A corrupted by a white Gaussien noise of power σ. Its periodogram can be written as : S x (n) = 1 N X(n) σ + A N 4 sin(π(k k 0 )) sin( π(k k0) N ) sin(π(k + k 0 )) + sin( π(k+k0) N ) (3) where X(n) represents the digital Fourier transform of the cosine denoted x. 5