inverse dft. frequency to time domain sometimes calculations are easier in the frequency domain then...
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Inverse DFT
Frequency to time domain
• Sometimes calculations are easier in the frequency domain then later convert the results back to the time domain
• Convert Time -> Frequency with DFT
• Convert Frequency -> Time with the Inverse Discrete Fourier Transform
• From Last week, the DFT is:
21
0
( ) ( )j hnNN
n
F h x n e
• The IDFT is:
21
0
1( ) ( )
j hnNN
h
x n F h eN
Where x is effectively a row matrix of size Nh is the required harmonicN is number of Fourier coefficientsF(h) is the complex DFT value
• To speed up the manual analysis, remember:
0
2
3
2
2
cos( ) sin( )
cos(0) sin(0) 1
cos( ) sin( )2 2
cos( ) sin( ) 1
3 3cos( ) sin( )
2 2
cos(2 ) sin(2 ) 1
j
j
j
j
j
j
e j
e j
e j j
e j
e j j
e j
• Relate this to the argand diagram…
• Similarly
0
2
3
2
2
cos( ) sin( )
cos(0) sin(0) 1
cos( ) sin( )2 2
cos( ) sin( ) 1
3 3cos( ) sin( )
2 2
cos(2 ) sin(2 ) 1
j
j
j
j
j
j
e j
e j
e j j
e j
e j j
e j
• So the vector rotates clockwise
Example• Consider the 4 DFT values generated from last
week’s example: {2,1+j,0,1-j}
21
0
2 01
0
2 1 2 0 2 1 2 334 4 4
0
3 3
2 2 2 2
1( ) (0) ( ) for 4
1 2 (1 ) 0 (1 )(0) (0) 1
4 4
1 1(1) (1) 2 (1 ) 0 (1 )
4 4
1 22 0
4
j hnNN
h
j hNN
h
j j j j
N
h
j j j j
x nT x X k e NN
j jx F e
x F e e j e j e
j j je je e je
00
4
Show that the other 2 terms for k=2 and k=3 are 0 and 1
j j j
DFT processing cost
• DFT processing cost is expensive– Each term is a product of a complex number– Each term is added so for an 8 point DFT
need 8 multiplies and 7 adds (N and N-1)– There are 8 harmonic components to be
evaluated (h=0 to 7)– So an 8 point DFT requires 8x8 complex
multiplications and 8x7 complex additions– An N point transform needs N2 Complex
multiplications and N(N-1) complex adds
Fast Fourier Transform
• Processing cost for DFT is:
2 2log Complex multiplications log Complex additions2
NN N N
2 Complex multiplications ( 1) Complex additionsN N N
• Processing cost for FFT is:
• 1024 point:
DFT: 1048576x and 1047552+
FFT: 5120x and 10240+