Chapter 1 -Discrete Signals A Sampled or discrete time signal x[n] is just an ordered sequence of values corresponding to the index n that embodies the time history of the signal A discrete signal is represented by a sequence of values x[n] ={1,2,3,4,5,….} The bar under 3 indicate that 3 is the center the origin where n=0; … denote infinite extant on that side. Discrete signal can be the left sided, right sided causal or anti causal.

Discrete pulse signal -rectangle Rect(n/2N) N-N

Periodicity for discrete signal The period of discrete signal is measured as the number of sample per period. x[n]=x[n±kN] k=0,1,2,3,4,…. The period N is the smallest number of sample s that repeats. N: always an Integer For combination of 2 or 3 or more signals N is the LCM (Least Common Multiple) of individual periods.

Fractional delays : Fractional delay of x[n] requires Interpolation, Shift and decimation each operation involve integers. In general for fractional delay we have the form of x[n-M/N]=x[(Nn-M)/N] we should do the following in order to get the answer of this delay first interpolate by N then delay by M and finally decimate by N. Example : x[n]={ 2,4,6,8 } find y[n]=x[n-0.5] assuming linear interpolation. y[n]=x[(2n-1)/2]. We first interpolate by 2 then shift by 1 and finally decimate by 2. After interpolation we get x[n/2]={2,3,4,5, 6,7,8,4}. Shift by 1 we obtain x2[n]={2,3,4, 5,6,7,8,4}; after decimation we get y[n]=x[(2n-1)/2]={3, 5,7,4};

Common discrete signal Dirac or delta Step function Ramp function r[n]=nu[n]

Step function …………………….. U[n] dirac  [n] Ramp function …………………..

Rect(n/2N) -N N

Tri(n/N) n Discrete Sinc Sinc(n/N)=sin(nπ/N)/ (nπ/N) Discrete Exponential : -N

Periodicity for discrete signal The period of discrete signal is measured as the number of sample per period. x[n]=x[n±kN] k=0,1,2,3,4,…. The period N is the smallest number of sample s that repeats. N: always an Integer For combination of 2 or 3 or more signals N is the LCM (Least Common Multiple) of individual periods.

Signal Measures : Summation of discrete signal discrete Sum Absolute sum Average of D.S Energy of D.S.

Power of periodic signal Operation on discrete Signals Time Shift : Y[n]=x[n-  ] if  >0, delay of the signal by . The signal is shifted toward the right (shift right). If  <0 advance of the signal by  and the shift is toward the left. Folding : y[n]=x[-n]; Example : x[n]=[1,2,3,4]; y[n]= [4,3,2,1]

y[n]=x[-n-  ] could be obtained by 2 methods 1- first make delay by  we get x[n-  ] then fold we get y[n]. 2- first fold x[n] we get x[-n] then shift left by  we get x[-n-  ]. Symmetry of Discrete Signals -Even symmetry: xe[n]=xe[-n] -Odd Symmetry Xo[n]=-xo[-n] X[n]=xe[n]+xo[n] ; xe[n]=0.5x[n]+0.5x[-n] xo[n]=0.5x[n]-0.5x[-n]

Decimation and interpolation of D.S. Decimation by N : Keep every Nth sample, this lead to potential loss of information. Example of decimation by 2: X[n]={1,2,6,4,8} after decimation xd[n]=x[2n]={1,6,8} Interpolation of the D.S. by N: Insert N-1 new values after each sample. The new value may be zero or the previous value or we calculate it using alinear interpolation. Example : x[n]={1,6,8}; using zero interpolation we get xi[n]=x[n/2]=[1,0,6,0,8,0}; by using step interpolation (previous value) we get xi[n]=[1,1,6,6,8,8}; And finally by using linear interpolation we get xi[n]={1,3.5,6,7,8,4}. Ex: linear interpolation by 3 of the following signal : x[n]=[1,7,13] XI[n]=[1,3,5,7,9,11,13,8.6,4.3]

Fractional delays : Fractional delay of x[n] requires Interpolation, Shift and decimation each operation involve integers. In general for fractional delay we have the form of x[n-M/N]=x[(Nn-M)/N] we should do the following in order to get the answer of this delay first interpolate by N then delay by M and finally decimate by N. Example : x[n]={ 2,4,6,8 } find y[n]=x[n-0.5] assuming linear interpolation. y[n]=x[(2n-1)/2]. We first interpolate by 2 then shift by 1 and finally decimate by 2. After interpolation we get x[n/2]={2,3,4,5, 6,7,8,4}. Shift by 1 we obtain x2[n]={2,3,4, 5,6,7,8,4}; after decimation we get y[n]=x[(2n-1)/2]={3, 5,7,4};

Discrete time Harmonic and sinusoids If we sample an analog sinusoid x(t)=cos(2  f 0 t + θ) at interval ts The sampling rate S=1/ts sample/s we obtain the following signal : X[n]=cos(2  f nts + θ ) = cos(2  nf /S + θ ) = cos(2  nF + θ ) ; With f: analog frequency and F : Digital frequency (cycles/sample)  F  (digital radian frequency); Digital frequency is an analog frequency normalized by sampling Rate S. F (cycles/Sample) = f (cycles/second) / S (sample/second) Discrete time (DT) harmonic are not always periodic in time It is periodic if it digital frequency F can be expressed as ratio of integers F=K/N ; the period of the signal is N if K/N does not have common factor.

one period of the DT harmonic is obtained from K period of analog sinusoid. DT harmonics are always periodic in Frequency DT harmonic is always periodic in frequency but not always unique its frequency F and F±M are identical for integer M. F is unique only if it lies in the principal period If F >0.5 the unique frequency is Fa=F-M where M is chosen to unsure

Aliasing and Sampling Theorem How to sample an analog signal without loss of information ? For an analog signal band limited to fmax Hz. The sampling rate S must exceed 2 fmax S=2fmax defines the Nyquist rate; ts=1/2fmax defines the Nyquist interval. For an analog sinusoid : The Nyquist rate correspond to taking 2 samples per period. The Highest frequency cannot exceed half of the sampling or reconstruction rate. Upon sampling or reconstruction all frequencies must be brought into the principal period. Aliasing Occurs if the analog signal is sampled below the Nyquist rate. If S <2fmax, the reconstructed analog signal is aliased to a lower frequency fa. We find fa as fa=f0-MS, where M is the integer that places fa in the principal period :

Reconstruction using different sampling rate Aliased or reconstructed frequencies are always identified by their principal period. Sampling : unique digital frequencies always lies in the principal period Reconstruction at SR : Analog frequencies lie in the principal period The reconstructed frequency is fr=Sr.Fa fr=Sr. fa/S

Random signals Measures for random variable: Variance Example of probability density for random variables Uniform density : Gaussian Density :

Statistical estimation of discrete signal Average : Variance : Signal to Noise Ratio SNR :for a noisy signal x(t)=S(t)+A n(t) S(t): signal; An(t): noise with amplitude A For a noisy signal averaging different realization of the signal reduce the noise and then increase the signal to noise ratio