Signals and Systems - Electrical Engineering

Problems 629

Table 10.1DTFT of Common Signals and DTFT Properties

Discrete-Time Fourier Transforms

Discrete-Time Signal DTFTX(ejω), Periodic of Period 2 π

1. 
   

   δ[n] 1, −π≤ω < π

   
   


2. 
   

   A 2 πAδ(ω), −π≤ω < π

   
   


3. 
   

   ejω^0 n 2 πδ(w−ω 0 ), −π≤ω < π

   
   


4. 
   

   αnu[n], |α|< (^11) −α^1 e−jω, −π≤ω < π

   
   


5. 
   

   nαnu[n], |α|< 1 αe
   −jω
   ( 1 −αe−jω)^2 , −π≤ω < π

   
   


6. 
   

   cos(ω 0 n)u[n] π[δ(ω−ω 0 )+δ(ω+ω 0 )], −π≤ω < π

   
   


7. 
   

   sin(ω 0 n)u[n] −jπ[δ(ω−ω 0 )+δ(ω+ω 0 )], −π≤ω < π

   
   


8. 
   

   α|n|, |α|< (^11) − 2 α^1 cos−α(ω)^2 +α 2 , −π≤ω < π

   
   


9. 
   

   u[n+N/2]−u[n−N/2] sin(ω(sin(ω/N+ 21 )/)^2 ), −π≤ω < π

   
   


10. 
    

    αncos(ω 0 n)u[n]^1 −αcos(ω^0 )e
    −jω
    1 − 2 αcos(ω 0 )e−jω+α^2 e−^2 jω, −π≤ω < π

    
    


11. 
    

    αnsin(ω 0 n)u[n] αsin(ω^0 )e
    −jω
    1 − 2 αcos(ω 0 )e−jω+α^2 e−^2 jω, −π≤ω < π
    Properties of the DTFT
    Z-transform: x[n],X(z),|z|= 1 ∈ROC X(ejω)=X(z)|z=ejω
    Periodicity: x[n] X(ejω)=X(ej(ω+^2 πk)), kinteger
    Linearity: αx[n]+βy[n] αX(ejω)+βY(ejω)
    Time-shift: x[n−N] e−jωNX(ejω)
    Frequency-shift: x[n]ejωon X(ej(ω−ω^0 ))
    Convolution: (x∗y)[n] X(ejω)Y(ejω)
    Multiplication: x[n]y[n] 21 π
    ∫π
    −πX(e
    jθ)Y(ej(ω−θ))dθ

    
    

Symmetry: x[n],real valued |X(ejω)|,even function ofω

∠X(ejω),odd function ofω

Parseval’s relation:

∑∞

n=∞|x[n]|

(^2) = 1
2 π
∫π
−π|X(e
jω| (^2) dω

Problems............................................................................................

10.1. Eigenfunction property and frequency response—MATLAB
An IIR filter is characterized by the difference equation

y[n]=0.5y[n−1]+x[n]− 2 x[n−1] n≥ 0

wherex[n]is the input andy[n]is the output of the filter. LetH(z)be the transfer function of the filter.