Digital Signal Processing/Transforms
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This page lists some of the transforms from the book, explains their uses, and lists some transform pairs of common functions.
Contents
ContinuousTime Fourier Transform (CTFT)[edit]
[CTFT]
CTFT Table[edit]
Time Domain  Frequency Domain  

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Notes: 


DiscreteTime Fourier Transform (DTFT)[edit]
DTFT Table[edit]
The information in the table below may be inaccurate  The DTFT transfers into the periodic frequency domain. The signals shown in the table are not periodic, and hence they are obviously wrong. The table seems to list ordinary Fourier transforms for angular frequency, which coincidentally also uses the greek letter ω, although with a different meaning.
DTFT Properties[edit]
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 Table[edit]
TimeDomain x[n] 
Frequency Domain X[k] 
Notes 

DFT Definition  
Shift theorem  
Real DFT  
ZTransform[edit]
ZTransform Table[edit]
Here:
 for , for
 for , otherwise
Signal,  Ztransform,  ROC  

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Bilinear Transform[edit]
see [1]