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:




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.,










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






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






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


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




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