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]}