Digital Signal Processing/Transforms

This page lists some of the transforms from the book, explains their uses, and lists some transform pairs of common functions.

Continuous-Time Fourier Transform (CTFT)Edit


[CTFT]

 

CTFT TableEdit

  Time Domain Frequency Domain
   
1    
2    
3    
4    
5    
6    
7    
8    
9    
10    
11    
12    
13    
14    
15    
16    
Notes:
  1.  
  2.   is the rectangular pulse function of width  
  3.   is the Heaviside step function
  4.   is the Dirac delta function

Discrete-Time Fourier Transform (DTFT)Edit

DTFT TableEdit

Time domain
  where  
Frequency domain
  where  
Remarks
    Definition
   
   
    Here   represents the delta function
which is 1 if   and zero otherwise.
   
      is 2π periodic
   

DTFT PropertiesEdit

Property Time domain
 
Frequency domain
 
Remarks
Linearity    
Shift in time     integer k
Shift in frequency     real number a
Time reversal    
Time conjugation    
Time reversal & conjugation    
Derivative in frequency    
Integral in frequency    
Convolve in time    
Multiply in time    
Correlation    

Where:

  •   is the convolution between two signals
  •   is the complex conjugate of the function x[n]
  •   represents the correlation between x[n] and y[n].

Discrete Fourier Transform (DFT)Edit

DFT TableEdit

Time-Domain
x[n]
Frequency Domain
X[k]
Notes
    DFT Definition
    Shift theorem
   
    Real DFT
     
     

Z-TransformEdit

Z-Transform TableEdit

Here:

  •   for  ,   for  
  •   for  ,   otherwise
Signal,   Z-transform,   ROC
1      
2      
3      
4      
5      
6      
7      
8      
9      
10      
11      
12      
13      
14      
15      
16      
17      
18      
19      
20      

Bilinear TransformEdit

see [1]

Discrete Cosine Transform (DCT)Edit

Haar TransformEdit