The Sampling/Sinc Function -6π -4π -2π 0 2π 4π 6π 1.0 0.8 0.6 0.4 sin(x)/x sin(πx)/(πx) 0.2 0.0 -0.2 -15-10 -5 0 5 10 15
The Sampling/Sinc Function
The Discrete-Time Unit Impulse Function ·Definition 1,n=0 6l=50.n≠0 Aka.Kronecker delta function
The Discrete-Time Unit Impulse Function • Definition 0, 0 1, 0 [ ] n n n • Aka, Kronecker delta function
Discrete-time form of impulse. 6[n] 1.09 00 n -4-3-2-1 0 12 3 4
Discrete-time form of impulse
Sampling Property of Unit Impulse The unit impulse can be used to sample the value of a signal at n=0 x[n]o[n]=x[0]8[n] ·More generally x[n]o[n-no]=xInololn -nol
Sampling Property of Unit Impulse • The unit impulse can be used to sample the value of a signal at n=0 x[n][n] x[0][n] • More generally [ ] [ ] [ ] [ ] 0 n0 n n0 x n n n x
Relationship between Step and Impulse The discrete-time unit impulse is the first difference of the discrete-time unit step The discrete-time unit step is the running sum of the discrete unit impulse (or,the superposition of delayed discrete-time unit impulses) [n]=w[n]-w[n-l] 4[n]=∑[m]=∑6[n-k] m=-00 k=0
Relationship between Step and Impulse 0 [ ] [ ] [ ] [ ] [ ] [ 1] k n m u n m n k n u n u n • The discrete-time unit impulse is the first difference of the discrete-time unit step • The discrete-time unit step is the running sum of the discrete unit impulse (or, the superposition of delayed discrete-time unit impulses)