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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)

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[CTFT]

CTFT Table

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  Time Domain Frequency Domain
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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)

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DTFT Table

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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

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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)

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DFT Table

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

Z-Transform

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Z-Transform Table

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Here:

  • for , for
  • for , otherwise
Signal, Z-transform, ROC
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2
3
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11
12
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17
18
19
20

Bilinear Transform

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see [1]

Discrete Cosine Transform (DCT)

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Haar Transform

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