Lecture 22: Reconstruction and Admissibility

Transcription

1 WAVELETS AND MULTIRATE DIGITAL SIGNAL PROCESSING Lecture 22: Reconstruction and Admissibility Prof.V.M.Gadre, EE, IIT Bombay Tutorials Q 1. Construct the STFT, CWT of the signal x(t) using Matlab and discuss the observations. cos(2π10t)when 0 t 5 cos(2π25t)when 5 t 10 x(t) = cos(2π50t)when 10 t 15 cos(2π100t)when 15 t 20 Ans. The STFT is one of the most straightforward approaches for performing time-frequency analysis and can help you easily understand the concept of time-frequency analysis. Short Time Fourier Transform (STFT) of the signal x(t) is computed using the hamming window as the window function v(t). Hamming window of length 0.1 Sec (100 sample point) is shown in the Figure 1 Figure 1: Hamming window of length 100 The color bar on the right of the pictures represent the coefficient values in the figure. Higher coefficient values have the color of dark red and lower coefficient values have the color of dark blue. It can be observed from the figures that a window of smaller length provides better resolution in time (Signal representation is well confined in time i.e no blurring across time) but poor resolution in frequency (Signal representation is not well confined in frequncy i.e. blurring 22-1

2 Figure 2: STFT of the signal x(t) with window length 0.1 Sec Figure 3: STFT of the signal x(t) with window length 0.25 Sec Figure 4: STFT of the signal x(t) with window length 0.5 Sec 22-2

3 Figure 5: STFT of the signal x(t) with window length 1 Sec across frequency). Similarly, a window of larger length provides poor resolution in time (Signal representation is not well confined in time i.e blurring across time) but better resolution in frequency ( Signal representation is well confined in frequncy i.e. no blurring across frequency). So, we can t obtain a fine time resolution and a fine frequency resolution simultaneously. By observing the figure 5, (window length=1sec), we can say the distinct frequency s in the signal x(t), because window of larger length provides provides better resolution in frequency. By observing the figure 1, (window length=0.1sec), we can say upto what time extent each distinct frequncy is present. Computing the CWT of the signal x(t). Consider the wavelet function ψ(t) as shown in the figure 6. Computing the coefficient values for different scales (s 0 =1 to 64) and for different shift s (τ) using the Matlab is shown in the figure 7. The color bar on the right of the pictures represent the coefficient values in the figure. Higher coefficient values have the color of dark red and lower coefficient values have the color of dark blue. It can be observed from the CWT that, in the time interval ]0,5[ most of the energy is concentrated for the scale around 40 i.e. s 0 =40. Hence in that time interval the signal has the lowest frequency components. (f 1 =10 Hz) and in the time interval ]15,20[ most of the energy is concentrated for the scale around 4 i.e s 0 =4. Hence in that time interval the signal has the highest frequncy components (f 4 =100 Hz) and the frequency is 10 times (Ratio of scales) the frequency of signal in the range ]0,5[ (i.e. f 4 =10*f 1 ) which is actually true. Similarly, by observing the scale for which coefficients have the maximum value, in a given time interval, we can calculate the frequency s present in that time interval. Matlab code for generating the STFT is given below. 22-3

4 Figure 6: Wavelet function ψ(t) Figure 7: CWT of signal x(t) 22-4

5 clear all; %sampling frequency fc=500; %duration of the signal T=20; %zero padding factor my_zero=10; %generate the signal t=linspace(0,t,fc*t); x=zeros(1,length(t)); %thresholds th1=0.25*t*fc; th2=0.5*t*fc; th3=0.75*t*fc; th4=t*fc; x(1:th1)=cos(2*pi*10*t(1:th1)); x((th1+1):th2)=cos(2*pi*25*t((th1+1):th2)); x((th2+1):th3)=cos(2*pi*50*t((th2+1):th3)); x((th3+1):th4)=cos(2*pi*100*t((th3+1):th4)); figure, plot(t,x); win_len=100; figure,spectrogram(x,win_len,0.2*win_len,win_len,fc); win_len=250; figure,spectrogram(x,win_len,0.2*win_len,win_len,fc); win_len=500; 22-5

6 figure,spectrogram(x,win_len,0.2*win_len,win_len,fc); win_len=1000; figure,spectrogram(x,win_len,0.2*win_len,win_len,fc); 22-6