1 / 37

DSP-Chapter 2

DSP-Chapter 2. Prepared by Deepak Kumar Rout. Chapter2. DISCRETE-TIME SIGNALS AND SYSTEMS. 2.0 Introduction 2.1 Discrete-Time Signals : Sequences 2.2 Discrete-Time Systems 2.3 Linear Time-Invariant Systems 2.4 Properties of Linear Time-Invariant Systems

wilforda
Download Presentation

DSP-Chapter 2

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. DSP-Chapter 2 • Prepared by • Deepak Kumar Rout prepared by:D K Rout

  2. Chapter2. DISCRETE-TIME SIGNALS AND SYSTEMS 2.0 Introduction 2.1 Discrete-Time Signals : Sequences 2.2 Discrete-Time Systems 2.3 Linear Time-Invariant Systems 2.4 Properties of Linear Time-Invariant Systems 2.5 Linear Constant-Coefficient Difference Equations 2.6 Frequency-Domain Representation 2.7 Representation of Sequences of the Fourier Transform 2.8 Symmetry Properties of the Fourier Transform 2.9 Fourier Transform Theorems prepared by:D K Rout

  3. 2.1.Discrete - Time Signals x[n]= x(t)|t=nT n : -1,0,1,2,… T: sampling period x(t) : analog signal ii) unit step sequence u[n] = 1, n0 0, n0 i) unit impulse signal(sequence) d[n] = 1, n=0 0, n0 prepared by:D K Rout

  4. iii) exponential/sinusoidal sequence x[n]= Aej(won+), Acos(won+) - not necessarily periodic in n with period 2p/wo - periodic in nwithperiod N (discrete number) for woN=2pk or wo= 2pk/N [note] x(t)= Ae j(wo t+) is periodic in t with period T= 2p/wo (continuous value) iv) general expression x[n] = Sx[k]d[n-k] prepared by:D K Rout

  5. x[n] y[n] T[ ] 2.2Discrete-Time Systems System : signal processor i) memoryless or with memory y[n] = f(x[n]), y[n]=f(x[n-k]) with delay ii) linearity x1[n]y1[n] x2[n]y2[n] a1x1[n] + a2x2[n] a1y1[n] + a2 y2[n] - e.g. T[a1x1[n] + a2x2[n]] = T[a1x1[n]] + T[a2x2[n]] = a1T[x1[n]] + a2T[x2[n]] prepared by:D K Rout

  6. iii) time-invariance x[n]  y[n]  x[n-n0]  y[n-n0] - e.g., T[x[n-n0]] = T[x[n]] | n  n-no - e.g., d[n]  h[n]  d[n-k]  h[n-k] - counter-example : decimator T[ ] = x[Mn] iv) causality y[n] for n=n1, depends on x[n] for nn1 only - counter-example : y[n] = x[n+1] - x[n] prepared by:D K Rout

  7. v) stability bounded input yields bounded output(BIBO) |x[n]| <  for all n  |y[n]| <  for all n - counter-example : y[n] = S u[k] = 0, n<0 n+1, n0 unbounded ( no fixed value By exists that keeps y[n]  By <  .) n k=- prepared by:D K Rout

  8. In general, let x[n] = Sx[k]d[n-k] In general, let x[n] = Sx[k]d[n-k] y[n] = T[ Sx[k]d[n-k]] k=- k 2.3 Linear Time-Invariant Systems d[n] h[n] LTI T[d[n]] : impulse response y[n] x[n] ( by linearity) = Sx[k]T[d[n-k]] k ( by time-invariance) = Sx[k]h[n-k] coefficient  = x[n]*h[n] = Sx[n-r]h[r] r=- Convolution! prepared by:D K Rout

  9. d[n] = 1, n=0 0, n0  Sh[k]x[n-k] k=0 In summary, h[n] x[n] y[n] y[n] = x[n]*h[n] LTI h[n] : unique characteristic of the LTI system - causal LTI system  y[n] = Sh[k]x[n-k] = k=- [note] h[n] = T[d[n]] = 0 n<0. as prepared by:D K Rout

  10. - Stable LTI System   S | x[n-k] | • | h[k] | Sh[k]x[n-k] |  |y[n]| = | k=- k=-  S| h[k] | <   By  Bx k=-  S| h[k] | Therefore, <  k=- In fact, this is necessary and sufficient condition for stability of a BIBO system. ( You prove it! )(*1) prepared by:D K Rout

  11. ~ ~ ~ x[n] y[n] y[n] - Example of non-LTI system - Decimator x[n] Decimator M y[n] = x[Mn] ? = x[n-1] = y[n-1] = x[M[n-1]] M=3 y[n] = x[Mn] prepared by:D K Rout

  12. ~ y[n] y[n] y[n-1]  No! = x[Mn-1]  x[M[n-1]] = y[n-1] prepared by:D K Rout

  13. 2.4 Properties of LTI System h[n] x[n] y[n] = x[n]*h[n] LTI i) parallel connection h[n] = h1[n] + h2[n] prepared by:D K Rout

  14. ii) cascade connection h[n] = h1[n]*h2[n] =h2[n]* h1[n] [note] distinctive feature of digital LTI system (*2) prepared by:D K Rout

  15. M M = Sbrx[n-r] = Sbrx[n-r] r = 0 r = 0 N Saky[n-k] k = 0 2.5 Linear Difference Equations LTI x[n] y[n] i) Case 1 : N=0  FIR System (set a0 =1, for convenience) y[n] For impulse input, x[n]=d[n], the response is h[n]= 0, n<0 or n>M br 0 n  M finite impulse response! prepared by:D K Rout

  16. M = Sbrx[n-r] r = 0 N Saky[n-k] k = 1 ii) Case 2 : N  0  IIR System (set a0 =1, for convenience) y[n] e.g., set N=1 (lst order), and a1 = -a  y[n] = b0 x[n] + ay[n-1] - For impulse input x[n] = d[n], the response is 1) If assume a causal system, i.e., y[n]=0 n<0. y[0] = b0 d[0] + ay[-1] = b0 y[1] = b0 d[1] + ay[0] = ab0 • • • y[n] = b0 d[n] + ay[n-1] = anb0 h[n] = anb0u[n] infinite impulse response! prepared by:D K Rout

  17. 2) If assume an anti-causal system, i.e., y[n]=0 n>0. y[n-1] = a-1(-b0 d[n] + y[n]) y[0] = a-1(-b0 d[1] + y[1]) = 0 y[-1] = a-1(-b0 d[0] + y[0]) = a-1b0 h[n] = -anb0u[-n-1] • • • y[-n] = a-1(- b0 d[n+1] + y[n+1]) = -anb0 prepared by:D K Rout

  18. ^ ^ y = Ax=lx  y[n] = Sh[k] ejw(n-k) ^ ^ x x k = - 2.6 Frequency-Domain Representation • Linear System x y = Ax A scalar, eigenvalue for eigenvector input • LTI System x[n] y[n]=x[n]*h[n] h[n] y[n]=ejwn*h[n] ejwn = H(ejw)ejwn Fourier transform  ( Sh[k] e-jwk )ejwn =H(ejw) ejwn = k = - prepared by:D K Rout

  19. Sh[k] e-jwk k = - p h[k] = 1/2p H(ejw) ejwkdw -p • Fourier Transform H(ejw) = You prove this! (*3) • Condition for existence of FT | X(ejw) | <   S | x[n] | <  “ absolutely summable” (BIBO stable condition) • Real - imaginary H(ejw) = HR(ejw) + j HI(ejw) prepared by:D K Rout

  20. • Magnitude-phase H(ejw) = | H(ejw) |ej H(ejw) (e.g.) ideal delay system x[n] y[n] = x[n-nd] h[n] y[n] = ejw(n-nd)= H(ejw) ejwn H(ejw) = e-jwnd ejwn HR(ejw) = coswnd HI(ejw) = - sinwnd | H(ejw) | = 1  H(ejw) = -wnd prepared by:D K Rout

  21. (e.g.) sinusoidal input x[n] = Acos(w0n + j) = (A/2) ejjejw0n + (A/2) e-jje-jw0n y[n] = H(ejw0) (A/2) ejjejw0n + H(e-jw0) (A/2) e-jje-jw0n = (A/2)(H(ejw0) ejjejw0n + H(e-jw0) e-jje-jw0n) y[n] = H(ejw) ejwn <1> = (A/2){ (H(ejw0) ejjejw0n ) + (H(ejw0) ejjejw0n)* } = ARe{H(ejw0) ejjejw0n} = A | H(ejw0) |(cosw0n + j + q) = A cos (w0(n-nd) + j) <2> <3> prepared by:D K Rout

  22. = ( Sh*[n] e-jw0n)* = H*(ejw0) n <1> if h[n] real h[n] = hR[n]+jhI[n] = hR[n] = h*[n] H(e-jw0) = Sh[n] e-(-jw0n) n <2> H(ejw) = | H(ejw) |ejq <3> if ideal delay system with | H(ejw) | = 1,  H(ejw) = q = -wnd prepared by:D K Rout

  23. (e.g.) ideal lowpass filter (LPF) x[n] y[n] 1 -wc wc Hl(ejw) = 1. e-jwnd|w|  0. elsewhere wc periodic with period 2p Input x[n] = Acos(w0n + j) output y[n] = Acos(w0(n-nd )+ j) , if 0 , otherwise wo < wc prepared by:D K Rout

  24. sinw0[n-nd ] p[n-nd ] wc hl [k] = e-jwnd ejwndw -wc (e.g.) Fourier transform of anu[n] |a|<1 X(ejw)   = Sane-jwn = S (ae-jw)n = n = 0 n = 0 (e.g.) inverse Fourier transform of ideal LPF = - <n <  prepared by:D K Rout

  25. - Data length vs. Spectrum Change - Gibb’s phenomenon (page 52) prepared by:D K Rout

  26. - Error reduces in RMS sense but not in Chebyshev sense.  Limitation of rectangular windowing prepared by:D K Rout

  27. 2.8 Symmetry Properties (table 2.1) i) even / odd e : conjugate symmetric  even o : conjugate anti-symmetric  odd prepared by:D K Rout

  28. ii) real/imaginary iii) conjugation/reversal S = ( S )* prepared by:D K Rout

  29. iv) real/imaginary - even/odd v) for real x[n] real  even imaginary  odd prepared by:D K Rout

  30. (e.g.) x[n] = anu[n] |a|<1, real (example 2.25) even odd even odd prepared by:D K Rout

  31. Dashed line : a = 0.5 Solid line : a = 0.9 prepared by:D K Rout

  32. 2.9 Fourier Transform Theorems (table 2.2) i) linearity ii) time shifting iii) frequency shifting iv) time reversal prepared by:D K Rout

  33. v) differentiation in frequency vi) Parserval’s relation vii) convolution relation prepared by:D K Rout

  34. viii) modulation/windowing relation ix) fundamental functions prepared by:D K Rout

  35. 1. 0. prepared by:D K Rout

  36. H.W. of Chapter 2 Matlab: [1] Consider the following discrete-time systems characterized by the difference equations: y[n]=0.5x[n]+0.27x[n-1]+0.77x[n-2] Write a MATLAB program to compute the output of the above systems for an input x[n]=cos(20n/256)+cos(200n/256), with 0n<299 and plot the output. [2] The Matlab command y=impz(num,den,N) can be used to compute the first N samples of the impulse response of the causal LTI discrete-time system. Compute the impulse response of the system described by y[n]-0.4y[n-1]+0.75y[n-2] =2.2403x[n]+2.4908x[n-1]+2.2403x[n-2] and plot the output using the stem function. prepared by:D K Rout

  37. H.W. of Chapter 2 Text: [3]2-15 [4]2-30 [5]2-42 [6]2-56 [7]2-58 prepared by:D K Rout

More Related