# Lecture 9: The DFT

We have seen that the Fourier transform can be used in a variety of applications. While an important theoretical tool, there is a problem with it in practice: we must have an analytical expression for the functions we are transforming, and have to be able to compute its transform. In practice, the transforms are often computed from discrete samples of a signal, and the transform is computed for only a discrete set of frequencies. This computational approach is known as the DFT. There are fast algorithms for computing the DFT which are called the Fast Fourier Transform (FFT). The FFT is nothing more than a way of organizing the computations in the DFT to compute exactly the same thing, but with less processing for the computer. It is safe to say that the revolution in signal processing has its genesis in the discovery of the FFT algorithm.

We will begin my making the connection between continuous-time signals with the spectrum and the DFT signals in their discrete time and frequency domains. Let be timelimited to seconds, having spectrum . Then since is timelimited, it cannot be bandlimited (why?). Let denote the sampled signal with samples taken ever seconds, and let denote the spectrum of the sampled signal, the periodic extension of , repeating ever Hz. Because is not bandlimited, there must be some amount of aliasing.

Now let us imagine sampling in the frequency domain of
; this corresponds to creating a periodic repetition of
the sampled signal in the time domain. We let
be the period of
repetition in the time domain, corresponding to samples every
of the spectrum.

Let us determine how many samples are involved here. The number of
samples in the time domain (of each period) is

The number of samples in the frequency domain of each period is

The aliasing can be reduced by increasing the sampling frequency, but can never be entirely eliminated for a timelimited signal. (If we had started out with a bandlimited signal, then it would not have been timelimited, and we would have had overlapping in the time domain.

If, in addition, the signal
that we are dealing with is not, in
fact timelimited, but we must for reasons of practicality deal with a
time-limited version, we must perform some truncation. This
truncation leads to
**
spectral leakage
**
. This also causes aliasing
(creating higher spectral components than might have been in the
original signal.) The spectral leakage can be reduced by increasing
the window width (longer data set). This increases
, which in
turn reduces
. (Note that
determines the frequency
resolution.)

Also observe that by this sampling property, we are getting only a partial view of the spectrum. A true spectral peak might not lie right on one of the sample values. This can be mitigated by sampling more finely (decrease , meaning increase , meaning increase .)

Let
be a discrete-time sequence, possibly obtained from a
continuous-time signal by sampling, say
. Note that we
are using
as the discrete time index. Suppose that we have
samples of the signal, where
is some number that we choose. (As
a point for future reference, the number
is commonly chosen to
be a power of 2, since this works most efficiently for most FFT
routines.) Following the book's notation, let

That is, it is simply a scaled version of the time sample. Often this scaling is overlooked (or it is assumed that the sampling interval is normalized to one). Also, let

That is, it is a sample of the spectrum.

The DFT of the set of points
is given by

where

In the transform formula, the number must be an integer. It is an index to a frequency (as we will see later). The inverse DFT is

The transform pair is sometimes represented as

or

Note the interesting parallels between these transforms and the F.T.s we have already seen:

- In going from the time domain ( ) to the frequency domain ( ), the exponential term in the summation has a negative sign.
- In going from the frequency domain to the time domain, the exponential term has a positive sign, and the summation is multiplied by a normalizing factor, in this case.

*exact*.

Let us connect the definition given above with the conventional
Fourier transform (noting where approximations are made). The sampled
signal can be written as

(time limited). The Fourier transform of this signal is

If we

*neglect the aliasing*, over the interval by the sampling theorem we have

Hence

Sampling this now we obtain

Let ; then

By our definition, . Putting these definitions together we find

We conclude that, except for aliasing, the DFT represents a sampled version of the FT.

Taking the last expression for
, multiply by
and sum:

We find (how?) that

Hence the sum collapses, and we get the desired result.

The value
is said to provide information about the
th
frequency ``bin''. It corresponds to a Hertz frequency of

Take some examples:

(This is the

**frequency resolution**.)

(This is the maximum representable frequency.) The number of points in the DFT can be written as

where relates to the sampling rate in the frequency domain, and relates to the sampling rate in the time domain. Observe that to increase the frequency resolution, the number of points

**of data**must be increased. We can increase and improve the frequency resolution by increasing (taking more data), which amounts to taking samples closer together in frequency, , or by taking samples faster in time (increasing the sampling rate).

**
Note:
**
The book defined the transform in terms of
. In practice,
the FFT routines simply deal with the data
;
they don't worry about whether it is scaled or not. Since the effect of the scaling
by
is
to simply scale the transform, in applications it is not common to worry about
the scaling either.

Note that zero padding cannot replace any information that is not
already present in the signal. It simply refines the sampling of the
spectrum that is already present.