Digital Signal Processing Reference
In-Depth Information
switch between the time domain signal and the frequency
domain spectral representation. The DFT takes a complex signal
and decomposes it into a sum of different frequency cosine and
sine waves.
In order to appreciate what is happening, we are going to
examine a few simple examples. This will involve multiplying and
summing up complex numbers, which, while not difficult, can be
tedious. We will minimize the tedium by using a short transform
length, but it cannot really be avoided in order to understand the
DFT.
We will start with the definition of the Fourier transform. The
basic Fourier theory tells us that any signal can be represented at
a sum of various frequency components, with a particular gain
and phase for each component. The signal, f(x), is multiplied and
integrated by each possible frequency. Each frequency is repre-
sented by the complex exponential, which is just a complex
sinusoid. This is also known as the Euler equation:
e
j
u
¼
cos
ðuÞþ
jsin
ðuÞ
The Fourier transform is defined as:
þ
N
j
u
dx
dx
FðuÞ¼
f
ðxÞe
N
Let's see if we can start simplifying: we will decide to perform
the calculation over a finite length of sampled data signal “x”,
which contains N samples, rather than continuous signal f(x) C
i
.
This gets rid of the infinity, and makes this something we can
actually build.
FðuÞ¼
X
N
1
X
i
e
j
u
i
i
¼
0
Note that
u
is a continuous variable, which we evaluated over
a2
. Now we will be transforming
a sampled time-domain signal to a sampled frequency-domain
spectral plot. So rather than computing
p
interval, usually from
p
to
p
u
continuously from
p
to
p
, we will instead compute
u
at M equally spaced points over
an interval of 2
. To avoid aliasing in the frequency domain, we
must make M
N. The reverse transform is the IDFT (or IFFT),
which reconstructs the sampled time-domain signal of length N
and does not require more than N points in the frequency
domain. Therefore we will set N
¼
M, and the frequency domain
representation will have the same number of points as the time
p
