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.

Continuous-Time Fourier Transform (CTFT)[edit]


[CTFT]

CTFT Table[edit]

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

Discrete-Time Fourier Transform (DTFT)[edit]

DTFT Table[edit]

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

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]

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

Z-Transform[edit]

Z-Transform Table[edit]

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 Transform[edit]

see [1]

Discrete Cosine Transform (DCT)[edit]

Haar Transform[edit]