Digital Signal Processing Reference
In-Depth Information
Discrete Fourier Transform (DFT):
N1
X
j2nm=N
X[m] =
x[n]e
n=0
where m = 0::N1.
Inverse Discrete Fourier Transform (IDFT):
N1
X
1
N
X[m]e
j2nm=N
x[n] =
m=0
where n = 0::N1.
Let's start with an example signal, x =fx
0
;x
1
;x
2
;x
3
g. We can nd the DFT of
x using the general formulas above, and since we know that N = 4, we can replace
it in the formula.
X
3
j2nm=4
X[m] =
x[n]e
n=0
j20m=4
+ x
1
e
j21m=4
+ x
2
e
j22m=4
+ x
3
e
j23m=4
X[m] = x
0
e
We can then nd the inverse transform...
3
X
x[n] =
1
4
X[m]e
j2nm=4
m=0
x[n] =
1
4
(X[0]e
j2n0=4
+ X[1]e
j2n1=4
+ X[2]e
j2n2=4
+ X[3]e
j2n3=4
)
...and replace the X[m] terms with their equivalents.
x[n] =
1
j200=4
+ x
1
e
j210=4
+ x
2
e
j220=4
+ x
3
e
j230=4
)e
j2n0=4
+
4
((x
0
e
j201=4
+ x
1
e
j211=4
+ x
2
e
j221=4
+ x
3
e
j231=4
)e
j2n1=4
+
(x
0
e
j202=4
+ x
1
e
j212=4
+ x
2
e
j222=4
+ x
3
e
j232=4
)e
j2n2=4
+
(x
0
e
j203=4
+ x
1
e
j213=4
+ x
2
e
j223=4
+ x
3
e
j233=4
)e
j2n3=4
)
(x
0
e
