Properties of the DFT
3.4.1 Properties of the DFT
The DFT has properties similar to those of the CTFS such as the real convolution property (2.5.13) since the DFT regards a time-domain sample sequence x[n] as one period of its periodic extension (with period equal to the DFT size N ) that can be described by
(3.4.9) where ˜x N [n] is the periodic extension of x[n] with period N and r N [n] the rectan-
x [n] = ˜x N [n] r N [n]
gular pulse sequence with duration of n = 0 : N − 1:
[n] = u s [n] − u s [n − N ] = (3.4.10b)
0 elsewhere
150 3 Discrete-Time Fourier Analysis On the other hand, the DFT sequence is born from the samples of the DTFT and
thus it inherits most of the DTFT properties except that the real translation and convolution in the time domain are not linear, but circular as with the CTFS. There- fore we discuss only the two properties and summarize all the DFT properties in Appendix B.
3.4.1.1 Real/Complex Translation – Circular Shift in Time/Frequency
Figure 3.8(a), (b1), (b2), and (d) show a sequence x[n] of finite duration N = 4, its periodic extension ˜x N [n], its shifted version ˜x N [n−M] (M = 2), and its rectangular-
windowed version ˜x N [n − M]r N [n], respectively. You can see more clearly from Fig. 3.8(a), (c1), (c2), and (d) that ˜x N [n − M]r N [n] is the circular-shifted (rotated) version of x[n] where Fig. 3.8(c1) and (c2) show the circular representation of the finite-duration sequence x[n] and its shifted or rotated version displayed around the circle with a circumference of N points. With this visual concept of circular shift, we can write the (circular) time-shifting property of DFS and DFT as
N [n − M] (circular shift) ↔W N X ˜ (k) (3.4.11a) ˜x DFT
˜x DFS
Mk
N [n − M]r N [n] (one period of circular shift) ↔W N X (k) (3.4.11b) We can also apply the duality between the time and frequency domains to write
Mk
the (circular) frequency-shifting property of DFS/DFT: W DFS
(3.4.12a) W N −Ln
N −Ln ˜x N [n] ↔ ˜X(k − L) (circular shift)
x DFT [n] ↔ ˜X(k − L)r N [k] (one period of circular shift) (3.4.12b)
x [n ]
n = 2 n = 1 n = 0 0 n 1 N –1
Circular representation
n = N –1
(a) (c1) Periodic repetition x ~
Circular shift N [n ]
01 N –1 n = N –1
(b1) (c2) Linear shift
Linear representation
N [n –2 ] Taking one period
x N [n –2 ]r N [n ]
(Rectangular windowing)
01 N –1 n (b2)
01 N –1 n
(d) Fig. 3.8 Circular shift of a sequence by 2 samples
3.4 Discrete Fourier Transform (DFT) 151
3.4.1.2 Real/Complex Convolution – Circular Convolution in Time/Frequency
Let us consider two sequences x[n] and y[n], both of duration M ≤ N and each with DFT X (k) and Y (k), respectively. Also let us denote the periodic extensions of
the two sequences and their DFSs by ˜x[n], ˜y[n], ˜ X (k), and ˜ Y (k), respectively. Then we can write the product of the two DFSs ˜ X (k) and ˜ Y (k) as
X km ˜ (k) = DFS
N −1
N { ˜x [n]} =
˜x[m] W
m=0
Y ˜ (k) = DFS kr N { ˜y [n]} = ˜y [r ] W
N −1
r =0
;˜ k X (k) ˜ Y (m+r) (k) = ˜x[m] ˜y [r ] W (3.4.13)
and compute its IDFS (inverse discrete Fourier series) as
˜ (3.4.8) 1 N −1
IDFS N X (k) ˜ Y (k) =
= N −1 ˜x[m] ˜y [r ] k W (r −n+m)
= N −1 ˜x[m]
N −1
r =0 ˜y [r ]δ[r − (n − m)]
m=0
= N −1 m =0 ˜x[m] ˜y [n − m] = ˜x[n] ∗ ˜y [n]
where ∗ denotes the circular or periodic convolution sum with period N . This N implies the real convolution property of DFS/DFT as
↔ ˜X(k) ˜Y (k) ( ˜x[n] DFT ˜y [n]) r
˜x[n] DFS ∗ ˜y [n] (circular convolution) (3.4.15a)
N ∗ N [n] (one period of circular convolution) ↔ X(k) Y (k) (3.4.15b) In duality with this, we can interchange the time and frequency indices to write the
complex convolution property as DFS 1 N −1
˜x[n] ˜y [n] ↔
X ˜ (k) ∗ N Y ˜ (k) (circular convolution) N
X ˜ (i ) ˜ Y (k − i) =
i =0
(3.4.16a) DFT 1
x [n] y [n] ↔
X ˜ (k) N
∗ ˜ N Y (k) r N [k] (one period of circular convolution) (3.4.16b)
152 3 Discrete-Time Fourier Analysis Note that the shift of a periodic sequence along the time/frequency axis actually
causes a rotation, making the sequence values appear to wrap around from the beginning of the sequence to the end.