Any signal can be decomposed in terms of sinusoidal (or complex exponential) components. Thus the analysis of signals can be done by transforming time domain signals into frequency domain and vice-versa.

**FREQUENCY TRANSFORMATIONS**

**INTRODUCTION**

Any
signal can be decomposed in terms of sinusoidal (or complex exponential)
components. Thus the analysis of signals can be done by transforming time
domain signals into frequency domain and vice-versa. This transformation
between time and frequency domain is performed with the help of Fourier
Transform(FT) But still it is not convenient for computation by DSP processors
hence Discrete Fourier Transform(DFT) is used.

Time
domain analysis provides some information like amplitude at sampling instant
but does not convey frequency content & power, energy spectrum hence
frequency domain analysis is used.

For
Discrete time signals x(n) , Fourier Transform is denoted as x(ω) & given
by

**DIFFERENCE BETWEEN FT & DFT**

**CALCULATION OF DFT & IDFT**

For
calculation of DFT & IDFT two different methods can be used. First method
is using mathematical equation & second method is 4 or 8 point DFT. If x(n)
is the sequence of N samples then consider WN= e ^{– j2 ∏ / N} (twiddle
factor)

**Four POINT DFT ( 4-DFT)**

**EIGHT POINT DFT ( 8-DFT)**

Examples:

Q) Compute DFT of x(n) = {0,1,2,3} Ans:
x4=[6, -2+2j, -2, -2-2j ]

Q) Compute DFT of x(n) = {1,0,0,1} Ans:
x4=[2, 1+j, 0, 1-j ]

Q) Compute DFT of x(n) = {1,0,1,0} Ans:
x4=[2, 0, 2, 0 ]

Q) Compute IDFT of x(k) = {2, 1+j, 0, 1-j } Ans: x4=[1,0,0,1]

**DIFFERENCE BETWEEN DFT & IDFT**

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Digital Signal Processing : Frequency Transformations : Frequency Transformations |

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