Sampling and Discrete Time Discrete-Time Signal Description Sampling is the acquisition of the values of a continuous-time signal at discrete points in time. x t discrete-time signal. ( ) is a continuous-time signal, x n x n = x ( nt s ) where T s is the time between samples is a M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 1 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl Sampling and Discrete Time The general form of a periodic sinusoid with fundamental period N 0 is g n [ ] = Acos( πf 0 n +θ) or Acos( Ω 0 n +θ) Unlike a continuous-time sinusoid, a discrete-time sinusoid is not necessarily periodic. If a sinusoid has the form g[ n] = Acos( πf 0 n +θ) then F 0 must be a ratio of integers (a rational number) for g[ n] to be periodic. If F 0 is rational in the form q / N 0 and all common factors in q and N 0 have been cancelled, then the fundamental period of the sinusoid is N 0, not N 0 / q (unless q = 1). M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 3 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 4 Periodic Periodic An Aperiodic Sinusoid Periodic Aperiodic M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 5 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 6 1
Two sinusoids whose analytical expressions look different, g 1 [ n] = Acos( πf 01 n +θ) and g [ n] = Acos( πf 0 n +θ) may actually be the same. If F 0 = F 01 + m, where m is an integer then (because n is discrete time and therefore an integer), ( ) = Acos( πf 0 n +θ) (Example on next slide) Acos πf 01 n +θ M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 7 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 8 Exponentials Real α Complex α M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 9 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 10 The Unit Impulse Function [ ] = 1, n = 0 δ n 0, n 0 The discrete-time unit impulse (also known as the Kronecker delta function ) is a function in the ordinary sense (in contrast with the continuous-time unit impulse). It has a sampling property, Aδ [ n n 0 ]x[ n] = Ax n 0 n= but no scaling property. That is, δ n [ ] [ ] = δ [ an] for any non-zero, finite integer a. The Unit Sequence Function [ ] = 1, n 0 u n 0, n < 0 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 11 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 1
The Signum Function 1, n > 0 sgn n = 0, n = 0 1, n < 0 The Unit Ramp Function ramp n [ ] = n, n 0 0, n < 0 = n u n n [ ] = u[ m 1] m= M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 13 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 14 The Periodic Impulse Function Let g[ n] be graphically defined by δ N [ n] = δ n mn m= [ ] g[ n] = 0, n >15 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 15 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 16 Time shifting n n + n 0, n 0 an integer Time compression n Kn K an integer > 1 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 17 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 18 3
Time expansion n n / K, K > 1 For all n such that n / K is an integer, g[ n / K ] is defined. For all n such that n / K is not an integer, g[ n / K ] is not defined. There is a form of time expansion that is useful. Let [ ] = x [ n / m ], n / m an integer y n 0, otherwise All values of y are defined. This type of time expansion is actually used in some digital signal processing operations. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 19 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 0 Differencing Accumulation [ ] = h[ m] g n n m= M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 1 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl Even and Odd Signals g[ n] = g[ n] g[ n] = g[ n] Products of Even and Odd Functions Two Even Functions g e [ ] + g[ n] [ n] = g n g o [ ] g[ n] [ n] = g n M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 3 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 4 4
Products of Even and Odd Functions An Even Function and an Odd Function Products of Even and Odd Functions Two Odd Functions M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 5 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 6 Symmetric Finite Summation Periodic Functions A periodic function is one that is invariant to the change of variable n n + mn where N is a period of the function and m is any integer. The minimum positive integer value of N for which [ ] = g[ n + N ] is called the fundamental period N 0. g n N g[ n] = g 0 n= N N g[ n] = 0 [ ] + g[ n] n=1 N n= N M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 7 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 8 Periodic Functions The signal energy of a signal x[ n] is E x = n= x[ n] M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 9 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 30 5
x[ n] = 0, n > 31 Find the signal energy of n= [ ] = x n ( 5 / 3) n, 0 n < 8 0, otherwise 7 n=0 ( ) n ( ) 4 E x = x[ n] = 5 / 3 = 5 / 3 N 1 N, r = 1 Using r n = 1 r N n=0, r 1 1 r 8 ( )4 E x = 1 5 / 3 1 5 / 3 7 n=0 ( ) 4 1.871 10 6 n M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 31 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 3 Some signals have infinite signal energy. In that case It is usually more convenient to deal with average signal power. The average signal power of a signal x n 1 P x = lim N N For a periodic signal x n P x = 1 N N 1 n= N x[ n] [ ] is [ ] the average signal power is n= N x[ n] The notation means the sum over any set of n= N consecutive n's exactly N in length. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 33 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 34 A signal with finite signal energy is called an energy signal. A signal with infinite signal energy and finite average signal power is called a power signal. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 35 6