# Fourier Transforms for Continuous/Discrete Time/Frequency

The Fourier transform can be defined for signals which are

- discrete or continuous in time, and
- finite or infinite in duration.

- discrete or continuous in frequency, and
- finite or infinite in bandwidth.

This book has been concerned almost exclusively with the discrete-time, discrete-frequency case (the DFT), and in that case, both the time and frequency axes are finite in length. In the following sections, we briefly summarize the other three cases. Table B.1 summarizes all four Fourier-transform cases corresponding to discrete or continuous time and/or frequency.

## Discrete Time Fourier Transform (DTFT)

The *Discrete Time Fourier Transform* (DTFT) can be viewed as the
limiting form of the DFT when its length is allowed to approach
infinity:

*continuous*normalized radian frequency variable,

^{B.1}and is the signal amplitude at sample number .

The inverse DTFT is

Instead of operating on sampled signals of length (like the DFT),
the DTFT operates on sampled signals defined over all integers
. As a result, the DTFT frequencies form a
*continuum*. That is, the DTFT is a function of
*continuous* frequency
, while the DFT is a
function of discrete frequency ,
. The DFT
frequencies
,
, are given by
the angles of points uniformly distributed along the unit circle
in the complex plane (see
Fig.6.1). Thus, as
, a continuous frequency axis
must result in the limit along the unit circle in the plane. The
axis is still finite in length, however, because the time domain
remains sampled.

## Fourier Transform (FT) and Inverse

The *Fourier transform* of a signal
,
, is defined as

and its inverse is given by

### Existence of the Fourier Transform

Conditions for the *existence* of the Fourier transform are
complicated to state in general [12], but it is *sufficient*
for to be *absolutely integrable*, *i.e.*,

*square integrable*,

*i.e.*,

There is never a question of existence, of course, for Fourier
transforms of real-world signals encountered in practice. However,
*idealized* signals, such as sinusoids that go on forever in
time, do pose normalization difficulties. In practical engineering
analysis, these difficulties are resolved using Dirac's ``generalized
functions'' such as the *impulse* (also called the
*delta function*) [38].

### The Continuous-Time Impulse

An *impulse* in continuous time must have *``zero width''*
and *unit area* under it. One definition is

An impulse can be similarly defined as the limit of

*any*integrable pulse shape which maintains unit area and approaches zero width at time 0. As a result, the impulse under every definition has the so-called

*sifting property*under integration,

provided is continuous at . This is often taken as the

*defining property*of an impulse, allowing it to be defined in terms of non-vanishing function limits such as

An impulse is not a function in the usual sense, so it is called
instead a *distribution* or *generalized function*
[12,38]. (It is still commonly called a ``delta function'',
however, despite the misnomer.)

## Fourier Series (FS) and Relation to DFT

In continuous time, a *periodic* signal , with period
seconds,^{B.2} may be expanded
into a *Fourier series* with coefficients given by

where is the th harmonic frequency (rad/sec). The generally complex value is called the th

*Fourier series coefficient*. The normalization by is optional, but often included to make the Fourier series coefficients independent of the fundamental frequency , and thereby depend only on the

*shape*of one period of the time waveform.

### Relation of the DFT to Fourier Series

We now show that the DFT of a sampled signal (of length ),
is proportional to the
*Fourier series coefficients* of the continuous
periodic signal obtained by
repeating and interpolating . More precisely, the DFT of the
samples comprising one period equals times the Fourier series
coefficients. To avoid aliasing upon sampling, the continuous-time
signal must be bandlimited to less than half the sampling
rate (see Appendix D); this implies that at most
complex harmonic components can be nonzero in the original
continuous-time signal.

If is bandlimited to
, it can be sampled
at intervals of seconds without aliasing (see
§D.2). One way to sample a signal inside an integral
expression such as
Eq.(B.5) is to multiply it by a continuous-time *impulse train*

where is the continuous-time impulse signal defined in Eq.(B.3).

We wish to find the continuous-time Fourier series of the
*sampled* periodic signal . Thus, we replace in
Eq.(B.5) by

^{B.3}

If the sampling interval is chosen so that it divides the signal period , then the number of samples under the integral is an integer , and we obtain

where . Thus, for all at which the bandlimited periodic signal has a nonzero harmonic. When is odd, can be nonzero for , while for even, the maximum nonzero harmonic-number range is .

In summary,

**Next Section:**

Selected Continuous-Time Fourier Theorems

**Previous Section:**

Fast Fourier Transform (FFT) Algorithms