Discrete-Time Signals Time-Domain Representation Signals are represented as sequences of numbers, called samples Sample value of a typical signal or sequence denoted as xn with n being an integer in the range-∞≤n≤o In] defined only for integer values of n and undefined for non-integer values of n Discrete-time signal represented by x[n Copyright C 2001, S K Mitra
1 Copyright © 2001, S. K. Mitra Discrete-Time Signals: Time-Domain Representation • Signals are represented as sequences of numbers, called samples • Sample value of a typical signal or sequence denoted as x[n] with n being an integer in the range • x[n] defined only for integer values of n and undefined for non-integer values of n • Discrete-time signal represented by {x[n]} − n
Discrete-Time Signals Time-Domain Representation Discrete-time signal may also be written as a sequence of numbers inside braces {xrn]}={…,-0.2,2.2,1.0.2,-3.7,2.9,… In the above, x1=-0.2,x0]=2.2,x1=1.1 etc The arrow is placed under the sample at time index n=0 Copyright C 2001, S.K. Mitra
2 Copyright © 2001, S. K. Mitra Discrete-Time Signals: Time-Domain Representation • Discrete-time signal may also be written as a sequence of numbers inside braces: • In the above, etc. • The arrow is placed under the sample at time index n = 0 { [ ]} ={,− 0.2,2.2,1.1,0.2,−3.7,2.9,} x n x[−1] = −0.2, x[0] = 2.2, x[1] =1.1
Discrete-Time Signals. Time-Domain Representation Graphical representation of a discrete-time signal with real-valued samples x 3456789101112 10-98-7-6-5-4-3-2-10120 1314151617 Copyright C 2001, S.K. Mitra
3 Copyright © 2001, S. K. Mitra Discrete-Time Signals: Time-Domain Representation • Graphical representation of a discrete-time signal with real-valued samples
Discrete-Time Signals Time-Domain Representation In some applications, a discrete-time sequence xn may be generated by periodically sampling a continuous-time signal xa(t )at uniform intervals of time K al 3T -5T -T0 T (37 Copyright C 2001, S.K. Mitra
4 Copyright © 2001, S. K. Mitra Discrete-Time Signals: Time-Domain Representation • In some applications, a discrete-time sequence {x[n]} may be generated by periodically sampling a continuous-time signal at uniform intervals of time x (t) a
Discrete-Time Signals. Time-Domain Representation Block-diagram representation of the sampling process xa( t) x[n]=xa(t x(n t=nt C 2.-1.0 Copyright C 2001, S.K. Mitra
5 Copyright © 2001, S. K. Mitra • Block-diagram representation of the sampling process ( ) a x t . . Discrete-Time Signals: Time-Domain Representation [ ] ( ) ( ) a a t nT x n x t x nT = = = n = ,− 2,−1,0,1,
Discrete-Time Signals. Time-Domain Representation Here, the n-th sample is given by xn]=xa(O)2=nr=xa(n7),n=…,-2-101 The spacing t between two consecutive samples is called the sampling interval or sampling period Reciprocal of sampling interval T, denoted as FT, is called the sampling frequency Copyright C 2001, S.K. Mitra
6 Copyright © 2001, S. K. Mitra Discrete-Time Signals: Time-Domain Representation • Here, the n-th sample is given by • The spacing T between two consecutive samples is called the sampling interval or sampling period • Reciprocal of sampling interval T, denoted as , is called the sampling frequency: x[n] x (t) x (nT), = a t=nT = a n = ,− 2,−1,0,1, FT T FT 1 =
Discrete-Time Signals. Time-Domain Representation Unit of sampling frequency is cycles per second, or Hertz(hz), if T is in seconds Whether or not the sequence xn has been obtained by sampling, the quantity xn] is called the n-th sample of the sequence Cin] is a real sequence, if the n-th sample xn is real for all values of n Otherwise, x[n is a complex sequence Copyright C 2001, S K Mitra
7 Copyright © 2001, S. K. Mitra Discrete-Time Signals: Time-Domain Representation • Unit of sampling frequency is cycles per second, or Hertz (Hz), if T is in seconds • Whether or not the sequence {x[n]} has been obtained by sampling, the quantity x[n] is called the n-th sample of the sequence • {x[n]} is a real sequence, if the n-th sample x[n] is real for all values of n • Otherwise, {x[n]} is a complex sequence
Discrete-Time Signals. Time-Domain Representation A complex sequence x[n can be written as xn=rein+jiim[n whe ere XmIn an nd ximIn] are the real and imaginary parts ofxn The complex conjugate sequence ofin is given by ( x*[n]=xre[n]-jinm[n) Often the braces are ignored to denote a sequence if there is no ambiguity Copyright C 2001, S.K. Mitra
8 Copyright © 2001, S. K. Mitra Discrete-Time Signals: Time-Domain Representation • A complex sequence {x[n]} can be written as where and are the real and imaginary parts of x[n] • The complex conjugate sequence of {x[n]} is given by • Often the braces are ignored to denote a sequence if there is no ambiguity x [n] re x [n] im {x[n]} {x [n]} j{x [n]} = re + im {x*[n]} {x [n]} j{x [n]} = re − im
Discrete-Time Signals. Time-Domain Representation Example-x[n=cos0 25n is a real sequence fIn])=(e03n is a complex sequence We can write vln=cos0 3n+sino3ng coso3n+j sinO3n) where rez reln]=cos03n Vim[n=(sino 3n) Copyright C 2001, S.K. Mitra
9 Copyright © 2001, S. K. Mitra Discrete-Time Signals: Time-Domain Representation • Example - is a real sequence • is a complex sequence • We can write where {x[n]}={cos0.25n} { [ ]} { } j . n y n e 0 3 = {y[n]}={cos0.3n + jsin0.3n} ={cos0.3n}+ j{sin0.3n} {y [n]} {cos . n} re = 0 3 {y [n]} {sin . n} im = 0 3
Discrete-Time Signals. Time-Domain Representation Example [n]} icosO.3ny-jsin03n=e /0.3n is the complex conjugate sequence of vn That is n]}={y*[m 10 Copyright C 2001, S.K. Mitra
10 Copyright © 2001, S. K. Mitra Discrete-Time Signals: Time-Domain Representation • Example - is the complex conjugate sequence of {y[n]} • That is, { [ ]} {cos . } {sin . } { } j . n w n n j n e 0 3 0 3 0 3 − = − = {w[n]}={y *[n]}
