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 | edit source]
[CTFT]
CTFT Table[edit | edit source]
Time Domain | Frequency Domain | |||||||||||
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11 | ||||||||||||
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13 | ||||||||||||
14 | ||||||||||||
15 | ||||||||||||
16 | ||||||||||||
Notes: |
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Discrete-Time Fourier Transform (DTFT)[edit | edit source]
DTFT Table[edit | edit source]
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 | edit source]
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 | edit source]
DFT Table[edit | edit source]
Time-Domain x[n] |
Frequency Domain X[k] |
Notes |
---|---|---|
DFT Definition | ||
Shift theorem | ||
Real DFT | ||
Z-Transform[edit | edit source]
Z-Transform Table[edit | edit source]
Here:
- for , for
- for , otherwise
Signal, | Z-transform, | ROC | |
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1 | |||
2 | |||
3 | |||
4 | |||
5 | |||
6 | |||
7 | |||
8 | |||
9 | |||
10 | |||
11 | |||
12 | |||
13 | |||
14 | |||
15 | |||
16 | |||
17 | |||
18 | |||
19 | |||
20 |
Bilinear Transform[edit | edit source]
see [1]