82.1 Discrete-Time Signals Time-Domain Representation Here, n-th sample is given by xn]=x2(t)l=nr=xa(nT),n=…,-2,-1,0,1, The spacing t between two consecutive samples is called the sampling interval or sampling period Reciprocal of sampling interval t, denoted as Fr, is called the sampling frequency: FT=1/T
§2.1 Discrete-Time Signals: Time-Domain Representation • Here, n-th sample is given by x[n]=xa (t) | t=nT=xa (nT), n=…,-2,-1,0,1,… • 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: FT=1/T
82.1 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 Rxn is a real sequence, if the n-th sample xn is real for all values of n Otherwise xn is a complex sequence
§2.1 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 • {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
82.1 Discrete-Time Signals Time-Domain Representation A complex sequence xn can be written as xn=aren+ximiN where x and x: are the real and imaginary parts of xn The complex conjugate sequence of xn is given by x*n=xreln-ximin Often the braces are ignored to denote a sequence if there is no ambiguity
§2.1 Discrete-Time Signals: Time-Domain Representation • A complex sequence {x[n]} can be written as {x[n]}={xre[n]}+j{xim[n]} where xre and xim are the real and imaginary parts of x[n] • The complex conjugate sequence of {x[n]} is given by {x*[n]}={xre[n]} - j{xim [n]} • Often the braces are ignored to denote a sequence if there is no ambiguity
82.1 Discrete-Time Signals Time-Domain Representation Example- xn=cos0 25n is a real sequence yn]=ejU3n is a complex sequence · We can write Ry n1=cos03n +jsin03n) ={c003n}+j{sin03n} where yreln=(cos03n RymInsin03n
§2.1 Discrete-Time Signals: Time-Domain Representation • Example - {x[n]}={cos0.25n} is a real sequence {y[n]}={ej0.3n} is a complex sequence • We can write {y[n]}={cos0.3n + jsin0.3n} = {cos0.3n} + j{sin0.3n} where {yre[n]}={cos0.3n} {yim[n]}={sin0.3n}
82.1 Discrete-Time Signals Time-Domain Representation Two types of discrete-time signals Sampled-data signals in which samples are continuous-valued Digital signals in which samples are discrete-valued Signals in a practical digital signal processing system are digital signaIs obtained by quantizing the sample values either by rounding or truncation
§2.1 Discrete-Time Signals: Time-Domain Representation • Two types of discrete-time signals: - Sampled-data signals in which samples are continuous-valued - Digital signals in which samples are discrete-valued • Signals in a practical digital signal processing system are digital signals obtained by quantizing the sample values either by rounding or truncation