1. 8 DIGITAL SIGNAL SPECTRA
2018/11/24
8 DIGITAL SIGNAL SPECTRA
2018/11/24

2. Main Contents
Demonstrates how to calculate spectra for nonperiodic signals using DTFT
Introduces the discrete Fourier series(DFS)
Shows how to calculate spectra for periodic signals using DFS
Defines fundamental frequency and harmonic frequency

3. 8.1 THE MEANING OF THE SPECTRUM
The smooth transition in a signal come from its low frequency components.
Sharp edges and rapid changes come from its high frequency elements.

4. 8.2 NONPERIODIC DIGITAL SIGNALS
To calculate the spectra of nonperiodic digital signals using DTFT
Truncate:截断

5. FIGURE 8-4 Rectangular pulse for Example 8.1.

6. FIGURE 8-6 Spectrum of signal for Example 8.1.

7. FIGURE 8-7 Magnitude and phase spectra for Example 8.2.
Solution:

9. FIGURE 8-8 The vowel “eee” (2000 samples, fs = 8 kHz)for Example 8. 3
FIGURE The vowel “eee” (2000 samples, fs = 8 kHz)for Example 8.3. And its magnitude spectrum.
demo:
listenMusic.m

10. 8.2 PERIODIC DIGITAL SIGNALS
FIGURE Periodic digital signals.

11. The introduction of DFS

12. DFS
DFS:Discrete Fourier Series(离散傅里叶级数)

13. Fourier Coefficients

14. Fourier Coefficients

15. frequency

16. frequency

17. FIGURE 8-11 Comparing DTFTs and DFSs for digital signals.

18. Comparing DTFT and DFS for digital signals
ck is the sample of DTFT
ck OR X(k)
See P461 Figure 11.28

19. FIGURE 8-12 Magnitude spectrum for a periodic signal.

20. FIGURE 8-13 Digital signal for Example 8.4.

21. FIGURE 8-14 Spectrum of signal for Example 8.4.

22. FIGURE 8-14 Spectrum of signal for Example 8.4.

23. FIGURE 8-15 Digital sine wave for Example 8.5.
SOLUTION:

24. MATLAB realization clear; N=10; n=0:1:(N-1); x=sin(n*pi/5);
for k=0:1:(N-1)
ef=exp(-j*2*pi/N*k*n);
Ck(k+1)=sum(x.*ef);
end
k=0:(N-1);
subplot(2,1,1);
stem(k,abs(Ck)/N);
subplot(2,1,2);
stem(k,angle(Ck));

25. ?

26. FIGURE 8-16 Spectrum of sine wave for Example 8.5.

27. analysis

28. FIGURE 8-17 Spectrum of signal for Example 8.6.
SOLUTION:

29. FIGURE 8-17 Spectrum of signal for Example 8.6.

30. FIGURE 8-18 Artificial “eee” signal for Example 8.7.

31. MATLAB realization % exm_8_7.m fs=8000; n=0:(8000-1); f1=200; f2=400;
W1=2*pi/fs*f1;
W2=2*pi/fs*f2;
x=2/3*sin(W1*n)+1/6*sin(W2*n);
sound(x,fs);
X=fft(x,1024);
plot(abs(X(1:512)));

33. FIGURE 8-19 Magnitude spectrum of artificial “eee” sound for Example 8

34. Notice
Leakage
For period signal, the sample number must be the multiple of period of signal.
demo

35. summary DTFT:discrete time Fourier transform
DFS:discrete Fourier series
DFT:discrete Fourier transform
FFT:fast Fourier transform

36. The Relationship

37. Calculation of DFT