华南理工大学：《数字信号处理》（双语版） Chapter 3 Discrete-Time Signals

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-o≤n≤o x[n] defined only for integer values of n and undefined for non-integer values of n Discrete-time signal represented by xnl 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 xn]}={,-0.2,2.2,1.1,0.2,-3.7,2.9, In the above,x[-1]-=-0.2,x0]=2.2,x=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-5] 3456789101112 0-98-7-6-5-4-3-2-1012 1314151617 x[3] 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 xa(-57) 3T 5T-37T-70T (3T 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(ol t=nt 1 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 川]=x1(O)=mr=x(m7)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 T 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 x[n has been obtained by sampling, the quantity x[n] is called the n-th sample of the sequence ixn 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 x[n]=re[n]+jMim[ni where reIn and ximin are the real and imaginary parts of xin The complex conjugate sequence ofxn is given by x In=re[]-jximnI 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[]=cos0 25n) is a real sequence ln =(e/0. 3n) is a complex sequence We can write n= coso 3n+sino 3n icos.3n+j(sin. 3ng where reln=cos0 3ni Vin[n=(sino 3ng 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 iw[n])=(cos03n,-j(sin03n)=(e 103n) is the complex conjugate sequence of yn) That is {v[n]}={y*[n]} 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]}