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 Heavyside step function
  3. is the Dirac delta function

Discrete-Time 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.

Time domain
where
Frequency domain
where
Remarks
Here represents the delta function
which is 1 if and zero otherwise.
for some
For any real number . Here is a periodic Dirac delta function. That is
and informally if

Trapezoid signal.svg

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]