prepared by:D K Rout DSP-Chapter 2 Prepared by  Deepak Kumar Rout

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

prepared by:D K Rout iii) exponential/sinusoidal sequence x[n]= Ae j(   n+  ), Acos(   n+  ) - not necessarily periodic in n with period 2  /   - periodic in n  with  period N (discrete number) for    2  k or    2  k/N [note] x(t)= Ae j(   t  +  ) is periodic in t with period T= 2  /   (continuous value) iv) general expression x[n] =   x[k]  [n-k]

prepared by:D K Rout 2.2Discrete-Time Systems T[ ] x[n]y[n] System : signal processor i) memoryless or with memory y[n] = f(x[n]), y[n]=f(x[n-k]) with delay ii) linearity x 1 [n]  y 1 [n] x 2 [n]  y 2 [n] a 1 x 1 [n] + a 2 x 2 [n]  a 1 y 1 [n] + a 2 y 2 [n] - e.g. T[a 1 x 1 [n] + a 2 x 2 [n]] = T[a 1 x 1 [n]] + T[a 2 x 2 [n]] = a 1 T[x 1 [n]] + a 2 T[x 2 [n]]

prepared by:D K Rout iii) time-invariance x[n]  y[n]  x[n-n 0 ]  y[n-n 0 ] - e.g., T[x [n-n 0 ]] = T[x [n]] | n  n-no - e.g.,  [n]  h[n]   [n-k]  h[n-k] - counter-example : decimator T[ ] = x[Mn] iv) causality y[n] for n=n 1, depends on x[n] for n  n 1 only - counter-example : y[n] = x[n+1] - x[n]

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

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

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

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

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

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

prepared by:D K Rout 2.4 Properties of LTI System LTI x[n]y[n] = x[n]*h[n] h[n] i) parallel connection h[n] = h 1 [n] + h 2 [n]

prepared by:D K Rout ii) cascade connection h[n] = h 1 [n]*h 2 [n] =h 2 [n]* h 1 [n] [note] distinctive feature of digital LTI system (*2)

prepared by:D K Rout 2.5 Linear Difference Equations N   a k  y[n-k] k = 0 M   b r  x[n-r] r = 0 LTI x[n]y[n] i) Case 1 : N=0  FIR System (set a 0 =1, for convenience) y[n] For impulse input, x[n]=  [n], the response is h[n]=0,n M b r 0  n  M finite impulse response! M   b r  x[n-r] r = 0

prepared by:D K Rout ii) Case 2 : N  0  IIR System (set a 0 =1, for convenience) y[n] e.g., set N=1 (lst order), and a 1 = -a  y[n] = b 0 x[n] + ay[n-1] M   b r  x[n-r] r = 0 N   a k  y[n-k] k = 1 - For impulse input x[n] =  [n], the response is 1) If assume a causal system, i.e., y[n]=0  n<0. y[0] = b 0  [0] + ay[-1] = b 0 y[1] = b 0  [1] + ay[0] = ab 0 y[n] = b 0  [n] + ay[n-1] = a n b 0 h[n] = a n b 0 u[n] infinite impulse response!

prepared by:D K Rout 2) If assume an anti-causal system, i.e., y[n]=0  n>0. y[n-1] = a -1 (-b 0  [n] + y[n]) y[0] = a -1 (-b 0  [1] + y[1]) = 0 y[-1] = a -1 (-b 0  [0] + y[0]) = a -1 b 0 h[n] = -a n b 0 u[-n-1] y[-n] = a -1 (- b 0  [n+1] + y[n+1]) = -a n b 0

prepared by:D K Rout 2.6 Frequency-Domain Representation A x ^ xy = Ax ^ y = Ax= x ^ scalar, eigenvalue for eigenvector input x ^ LTI System h[n] x[n]y[n]=x[n]*h[n] ejnejn y[n]=e j  n *h[n] = H(e j  )e j  n Fourier transform   y[n] =   h[k] e j  (n-k) k = -    h[k] e -j  k )e j  n =H(e j  ) e j  n k = -   = Linear System

prepared by:D K Rout Fourier Transform H(e j  ) =   h[k] e -j  k k = -   h[k] = 1/2  H(e j  ) e j  k d    You prove this! (*3) Condition for existence of FT | X(e j  ) | <    | x[n] | <  “ absolutely summable” (BIBO stable condition) Real - imaginary H(e j  ) = H R (e j  ) + j H I (e j  )

prepared by:D K Rout Magnitude-phase H(e j  ) = | H(e j  ) |e j  H(e j  ) (e.g.) ideal delay system h[n] x[n]y[n] = x[n-n d ] ejnejn y[n] = e j  (n-n d ) = H(e j  ) e j  n H(e j  ) = e -j  n d H R (e j  ) = cos  n d H I (e j  ) = - sin  n d | H(e j  ) | = 1  H(e j  ) = -  n d

prepared by:D K Rout (e.g.) sinusoidal input x[n] = Acos(  0 n +  ) = (A/2) e j  e j  0 n + (A/2) e -j  e -j  0 n y[n] = H(e j  0 ) (A/2) e j  e j  0 n + H(e -j  0 ) (A/2) e -j  e -j  0 n = (A/2)(H(e j  0 ) e j  e j  0 n + H(e -j  0 ) e -j  e -j  0 n ) = (A/2){ (H(e j  0 ) e j  e j  0 n ) + (H(e j  0 ) e j  e j  0 n ) * } = A  Re{H(e j  0 ) e j  e j  0 n } = A | H(e j  0 ) |(cos  0 n +  +  ) = A cos (  0 (n-n d ) +  y[n] = H(e j  ) e j  n

prepared by:D K Rout H(e j  ) = | H(e j  ) |e j  if ideal delay system with | H(e j  ) | = 1,  H(e j  ) =  = -  n d if h[n] real h[n] = h R [n]+jh I [n] = h R [n] = h * [n] H(e -j  0 ) =   h[n] e -(-j   n) n   h * [n] e -j   n ) * = H * (e j  0 ) n

prepared by:D K Rout (e.g.) ideal lowpass filter (LPF) 1 -c-c cc x[n]y[n] H l (e j  ) = 1. e -j  n d |  |  0.elsewhere periodic with period 2  Inputx[n] = Acos(  0 n +  ) output y[n] = Acos(  0 (n-n d )+  ), if 0,otherwise oo <  c cc

prepared by:D K Rout (e.g.) Fourier transform of a n u[n]|a|<1 X(e j  ) n = 0   =   a n e -j  n =   ae -j  ) n  n = 0 = (e.g.) inverse Fourier transform of ideal LPF h l [k] = e -j  n d e j  n d  cc  c sin  0 [n-n d ]  [n-n d ] -  <n <  =

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

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

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

prepared by:D K Rout ii) real/imaginary iii) conjugation/reversal  

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

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

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

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

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

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

prepared by:D K Rout 1. 0.

prepared by:D K Rout 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] x[n-1] x[n-2] and plot the output using the stem function.

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