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.

Periodicity for discrete signal The period of discrete signal is measuredas 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]={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]=[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}.

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-0.5] assuming linear interpolation. 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]

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

Properties of the discrete impulses Signal representation by impulses Examples

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