Let's consider the Continuous-Time Fourier Transform (CTFT)
pair:
XΩ=∫−∞∞xte−(jΩt)d
t
X
Ω
t
x
t
Ω
t
xt=12π∫−∞∞XΩejΩtd
Ω
x
t
1
2
Ω
X
Ω
Ω
t
The Fourier transform pair supplies us with our notion of
"frequency." In other words, all of our intuitions regarding the
relationship between the time domain and the frequency domain
can be traced to this particular transform pair.
It will be useful to view the CTFT in terms of basis
elements. The inverse CTFT equation above says that the
time-domain signal
xt
x
t
can be expressed as a weighted
summation of basis elements
bΩt
bΩt
−∞<Ω<∞
bΩ
t
Ω
bΩ
t
, where
bΩt=ejΩt
bΩ
t
Ω
t
is the basis element corresponding to frequency
ΩΩ. In other words, the
basis elements are parameterized by the variable
ΩΩ that we call
frequency. Finally,
XΩ
X
Ω
specifies the weighting coefficient for
bΩt
bΩ
t
. In the case of the CTFT, the number of basis
elements is uncountably infinite, and thus we need an integral
to express the summation.
The Fourier Series (FS) can be considered as a special sub-case
of the CTFT that applies when the time-domain signal is
periodic. Recall that if
xt
x
t
is periodic with period
TT, then it can be expressed as a
weighted summation of basis elements
bk
t|k=−∞∞
k
bk
t
, where
bk
t=ej2πTtk
bk
t
2
T
t
k
:
xt=∑k=−∞∞Xkej2πTtk
x
t
k
X
k
2
T
t
k
Xk=1T∫−T2T2xte−(j2πTtk)dt
X
k
1
T
t
T
2
T
2
x
t
2
T
t
k
Here the basis elements comes from a countably-infinite set,
parameterized by the frequency index
k∈Z
k
. The coefficients
Xk|k=−∞∞
k
X
k
specify the strength of the corresponding basis
elements within signal
xt
x
t
.
Though quite popular, Fourier analysis is not always the best
tool to analyze a signal whose characteristics vary with
time. For example, consider a signal composed of a periodic
component plus a sharp "glitch" at time
t0
t0, illustrated in time- and frequency-domains, Figure 1.
Fourier analysis is successful in reducing the
complicated-looking periodic component into a few simple
parameters: the frequencies
Ω1
Ω2
Ω1
Ω2
and their corresponding magnitudes/phases. The glitch
component, described compactly in terms of the time-domain
location
t0t0
and amplitude, however, is not described efficiently in the
frequency domain since it produces a wide spread of frequency
components. Thus, neither time- nor frequency-domain
representations alone give an efficient description of the
glitched periodic signal: each representation distills only
certain aspects of the signal.
As another example, consider the linear chirp
xt=sinΩt2
x
t
Ω
t
2
illustrated in Figure 2.
Though written using the
sin·
·
function, the chirp is not described by a single
Fourier frequency. We might try to be clever and write
sinΩt2=sinΩt·t=sinΩt·t
Ω
t
2
Ω
t
·
t
Ω
t
·
t
where it now seems that signal has an instantaneous
frequency
Ωt=Ωt
Ω
t
Ω
t
which grows linearly in time. But here we must be
cautious! Our newly-defined instantaneous frequency
Ωt
Ω
t
is not consistent with the
Fourier notion of frequency. Recall that the CTFT says that a
signal can be constructed as a superposition of fixed-frequency
basis elements
ejΩt
Ω
t
with time support from
−∞
to
+∞
+
; these elements are evenly spread out over all time,
and so there is noting instantaneous about Fourier frequency!
So, while instantaneous frequency gives a compact
description of the linear chirp, Fourier analysis is not capable
of uncovering this simple structure.
As a third example, consider a sinusoid of frequency
Ω0Ω0
that is rectangularly windowed to extract only one period (Figure 3).
Instantaneous-frequency arguments would claim that
xt=sinΩt·t ,
Ωt={Ω0 if t∈window0 if t∉window
Ω
t
Ω0
t
window
0
t
window
x
t
Ω
t
·
t
where
Ωt
Ω
t
takes on exactly two distinct "frequency" values. In
contrast, Fourier theory says that rectangular windowing induces
a frequency-domain spreading by a
sinΩΩ
Ω
Ω
profile, resulting in a continuum of Fourier
frequency components. Here again we see that Fourier analysis
does not efficiently decompose signals whose "instantaneous
frequency" varies with time.
