Discrete-time Signals & Systems

1. Discrete-time Signals & Systems Discrete-Time Signals

2. The correct representation of a discrete-time signal in Matlab takes 2 vectors. • One vector is used to indicate the locations of the time samples. • The other vector is used to indicate the amplitude (value) of the signal at the corresponding temporal locations. How to represent a discrete-time signal in Matlab?

3. Unit sample sequence: δ(n) = 1, n = 0 = 0, n ≠ 0 Basic Signals

4. Unit step sequence: u(n) = 1, n ≥ 0 = 0, n < 0 Basic Signals

5. Real-valued exponential sequence: x(n) = an, a is a real number Basic Signals

6. Complex-valued exponential sequence: x(n) = e(σ + j ω) n Basic Signals

7. Sinusoidal sequence: X(n) = A cos(ω n + θ) Basic Signals

8. Random sequences: rand(1, N) Basic Signals

9. Periodic sequence: x(n) = x(n+N) the smallest integer N is the fundamental period Basic Signals

10. Signal addition: {x1(n)} + {x2(n)} ={x1(n)+x2(n)} Basic Operations

11. Signal multiplication {x1(n)} × {x2(n)} ={x1(n) × x2(n)} Basic Operations

12. Scaling: α {x(n)} ={α x(n)} Basic Operations

13. Shifting: y(n) = { x(n - k) } y(m + k) = { x(m) } Basic Operations

14. Folding: y(n) = {x(-n)} Basic Operations

15. Sample Summation: x(n1)+…+x(n2) = sum(x(n1:n2)) Basic Operations

16. Sample products x(n1) × … × x(n2) = prod(x(n1:n2)) Basic Operations

17. Signal energy: |x(n1)|2 + … + |x(n2)|2 = sum(abs(x).^2) Basic Operations

18. Signal power: Average power of a periodic signal with fundamental period N 1/N (|x(1)|2 +…+|x(N)|2) Basic Operations

19. Unit sample synthesis: Useful Results

20. Even and odd synthesis • Even signal: xe (-n) = xe (n) • Odd signal: xo (-n) = - xo(n) • x(n) = xe(n) + xo (n), xe(n) = ½ (x(n) + x(-n)) xo(n) = ½ (x(n) - x(-n)) Useful Results

21. The geometric series 1 + α + α2 + … + α∞ = 1/(1-α) for |α| < 1 1 + α + α2 + … + αN-1 = (1-αN)/(1-α) for any α Useful Results

22. Correlation of sequences: rx,y(m) = sum_n (x(n) y(n-m)) Useful Results

23. x(n) = 2δ(n+2) – δ(n-4), -5≤n≤5 • x(n)=n[u(n)-u(n-10)]+10e-0.3(n-10)[u(n-10)-u(n-20)], 0≤n≤20 • x(n)=cos(0.04πn)+0.2w(n), 0≤n≤50, where w(n) is a Gaussian random sequence with zero mean and unit variance • x(n)={…,5,4,3,2,1,5,4,3,2,1,5,4,3,2,1,…}; -10≤n≤9 Example 1

24. Let x(n) = {1,2,3,4,5,6,7,6,5,4,3,2,1}. Determine and plot the following sequences. • x1(n)=2x(n-5)-3x(n+4) • x2(n)=x(3-n)+x(n)x(n-2) Example 2

25. Generate the complex-valued signal x(n)=e(-0.1+j0.3)n, -10≤n≤10 And plot its magnitude, phase, the real part and the imaginary part in four separate subplots. Example 3

26. Let x(n)=u(n)-u(n-10). Decompose x(n) into even and odd components. Example 4

27. y(n) = T[x(n)] Discrete Systems

28. A discrete system L[] is linear, if and only if it satisfies the principle of superposition. L[a1x1(n)+a2x2(n)]=a1L[x1(n)] + a2L[x2(n)] Linear Discrete Systems

30. If y(n) = L[x(n)] then L[x(n-k)]=y(n-k) Linear time-invariant (LTI) system

31. Impulse Response

32. Convolution

33. { x(n); nxb≤ n ≤ nxe } and { h(n); nhb≤ n ≤ nhe} nyb = nxb+ nhb nye= nxe + nhe Convolution: Matlab Implementation

34. Correlation is convolution after folding.

35. x(n)=[3, 11, 7, 0, -1, 4, 2] • y(n)=x(n-2)+w(n), where w(n) is a sequence of random noise • Compute the cross-correlation between y(n) and x(n) Example 5