# Selected Continuous Fourier Theorems

This section presents continuous-time Fourier theorems that go beyond obvious analogs of the DTFT theorems proved in §2.3 above. The differentiation theorem comes up quite often, and its dual pertains as well to the DTFT. The scaling theorem provides an important basic insight into time-frequency duality. The Poisson Summation Formula (PSF) in continuous time extends the discrete-time version presented in §8.3.1. Finally, the extremely fundamental uncertainty principle is derived from the scaling theorem.

## Radians versus Cycles

Our usual frequency variable is
in *radians per second*.
However, certain Fourier theorems are undeniably simpler and more
elegant when the frequency variable is chosen to be
in
*cycles per second*. The two are of course related by

(B.1) |

As an example, is more compact than . On the other hand, it is nice to get rid of all normalization constants in the Fourier transform and its inverse:

(B.2) | |||

(B.3) |

The ``editorial policy'' for this book is this: Generally, is preferred, but is used when considerable simplification results.

## Differentiation Theorem

Let denote a function differentiable for all such that and the Fourier transforms (FT) of both and exist, where denotes the time derivative of . Then we have

(B.4) |

where denotes the Fourier transform of . In operator notation:

(B.5) |

*Proof: *
This follows immediately from integration by parts:

since .

## Differentiation Theorem Dual

**Theorem: **Let
denote a signal with Fourier transform
, and let

(B.6) |

denote the derivative of with respect to . Then we have

(B.7) |

where denotes the Fourier transform of .

*Proof: *
We can show this by direct differentiation of the definition of the
Fourier transform:

An alternate method of proof is given in §2.3.13.

The transform-pair may be alternately stated as follows:

(B.8) |

## Scaling Theorem

The *scaling theorem* (or *similarity theorem*) provides
that if you horizontally ``stretch'' a signal by the factor
in the time domain, you ``squeeze'' and amplify its Fourier transform
by the same factor in the frequency domain. This is an important
general Fourier duality relationship.

**Theorem: **For all continuous-time functions
possessing a Fourier
transform,

(B.9) |

where

(B.10) |

and is any nonzero real number (the abscissa stretch factor). A more commonly used notation is the following:

(B.11) |

*Proof: *
Taking the Fourier transform of the stretched signal gives

The absolute value appears above because, when , , which brings out a minus sign in front of the integral from to .

## Shift Theorem

The *shift theorem* for Fourier transforms states that
*delaying* a signal
by
seconds multiplies its Fourier
transform by
.

*Proof: *

Thus,

(B.12) |

## Modulation Theorem (Shift Theorem Dual)

The Fourier *dual* of the shift theorem is often called
the *modulation theorem*:

(B.13) |

This is proved in the same way as the shift theorem above by starting with the

*inverse*Fourier transform of the right-hand side:

or,

(B.14) |

## Convolution Theorem

The *convolution theorem* for Fourier transforms states that
*convolution in the time domain equals multiplication in the
frequency domain*. The continuous-time
*convolution* of two signals
and
is defined by

(B.15) |

The Fourier transform is then

or,

(B.16) |

Exercise:Show that

(B.17) |

when frequency-domain convolution is defined by

(B.18) |

where is in radians per second, and that

(B.19) |

when frequency-domain convolution is defined by

(B.20) |

with in Hertz.

## Flip Theorems

Let the *flip operator* be denoted by

where denotes time in seconds, and denotes frequency in radians per second. The following Fourier pairs are easily verified:

The proof of the first relation is as follows:

## Power Theorem

The *power theorem* for Fourier transforms states that the
*inner product* of two signals in the time domain equals
their inner product in the frequency domain.

The *inner product* of two spectra
and
may
be defined as

(B.21) |

This expression can be interpreted as the inverse Fourier transform of evaluated at :

(B.22) |

By the convolution theorem (§B.7) and flip theorem (§B.8),

(B.23) |

which at gives

(B.24) |

Thus,

(B.25) |

## 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*pulse shape which maintains unit area and approaches zero width at time 0 [150]. 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

(B.28) |

(Note, incidentally, that is in but not .)

An impulse is not a function in the usual sense, so it is called
instead a *distribution* or *generalized function*
[36,150]. (It is still commonly called a ``delta function'',
however, despite the misnomer.)

## Gaussian Pulse

The *Gaussian pulse* of width (second central moment)
centered on time 0 may be defined by

(B.29) |

where the normalization scale factor is chosen to give unit area under the pulse. Its Fourier transform is derived in Appendix D to be

(B.30) |

## Rectangular Pulse

The *rectangular pulse* of width
centered on time 0 may be
defined by

(B.31) |

Its Fourier transform is easily evaluated:

Thus, we have derived the Fourier pair

Note that

**sinc**is the Fourier transform of the one-second rectangular pulse:

sinc | (B.33) |

From this, the scaling theorem implies the more general case:

sinc | (B.34) |

## Sinc Impulse

The preceding Fourier pair can be used to show that

(B.35) |

*Proof: *The inverse Fourier transform of
**sinc**
is

In particular, in the middle of the rectangular pulse at , we have

(B.36) |

This establishes that the algebraic area under

**sinc**is 1 for every . Every delta function (impulse) must have this property.

We now show that
**sinc**
also satisfies the *sifting
property* in the limit as
. This property fully
establishes the limit as a valid impulse. That is, an impulse
is *any* function having the property that

(B.37) |

for every continuous function . In the present case, we need to show, specifically, that

(B.38) |

Define

**sinc**. Then by the power theorem (§B.9),

(B.39) |

Then as , the limit converges to the algebraic area under , which is as desired:

(B.40) |

We have thus established that

(B.41) |

where

sinc | (B.42) |

For related discussion, see [36, p. 127].

## Impulse Trains

The *impulse* signal
(defined in §B.10)
has a constant Fourier transform:

(B.43) |

An impulse

*train*can be defined as a sum of shifted impulses:

(B.44) |

Here, is the

*period*of the impulse train, in seconds--

*i.e.*, the

*spacing*between successive impulses. The -periodic impulse train can also be defined as

where is the so-called

*shah symbol*[23]:

(B.46) |

Note that the scaling by in (B.46) is necessary to maintain unit area under each impulse.

We will now show that

(B.47) |

That is, the Fourier transform of the normalized impulse train is exactly the same impulse train in the frequency domain, where denotes time in seconds and denotes frequency in Hz. By the scaling theorem (§B.4),

(B.48) |

so that the -periodic impulse-train defined in (B.46) transforms to

Thus, the -periodic impulse train transforms to a -periodic impulse train, in which each impulse contains area :

(B.49) |

*Proof: *
Let's set up a limiting construction by defining

(B.50) |

so that . We may interpret as a

*sampled rectangular pulse*of width seconds (yielding samples).By

*linearity*of the Fourier transform and the

*shift theorem*(§B.5), we readily obtain the transform of to be

Using the closed form of a geometric series,

(B.51) |

with , we can write this as

where we have used the definition of given in Eq. (3.5) of §3.1. As we would expect from basic sampling theory, the Fourier transform of the sampled rectangular pulse is an aliased sinc function. Figure 3.2 illustrates one period for .

The proof can be completed by expressing the aliased sinc function as a sum of regular sinc functions, and using linearity of the Fourier transform to distribute over the sum, converting each sinc function into an impulse, in the limit, by §B.13:

by §B.13. Note that near , we have

as , as shown in §B.13. Similarly, near , we have

(B.52) |

as . Finally, we expect that the limit for non-integer can be neglected since

(B.53) |

whenever and is some integer, as implied by §B.13.

See, *e.g.*, [23,79] for more about impulses
and their application in Fourier analysis and linear systems theory.

Exercise:Using a similar limiting construction as before,

(B.54) |

show that a direct inverse-Fourier transform calculation gives

(B.55) |

and verify that the peaks occur every seconds and reach height . Also show that the peak widths, measured between zero crossings, are , so that the area under each peak is of order 1 in the limit as . [Hint: The shift theorem forinverseFourier transforms is , and .]

## Poisson Summation Formula

As shown in §B.14 above, the Fourier transform of an impulse train is an impulse train with inversely proportional spacing:

(B.56) |

where

(B.57) |

Using this Fourier theorem, we can derive the continuous-time PSF using the

*convolution theorem*for Fourier transforms:

^{B.1}

(B.58) |

Using linearity and the shift theorem for

*inverse*Fourier transforms, the above relation yields

We have therefore shown

Compare this result to Eq. (8.30). The left-hand side of (B.60) can be interpreted ,

*i.e.*, the time-alias of on a block of length . The function is periodic with period seconds. The right-hand side of (B.60) can be interpreted as the inverse Fourier

*series*of

*sampled*at intervals of Hz. This sampling of in the frequency domain corresponds to the aliasing of in the time domain.

## Sampling Theory

The dual of the Poisson Summation Formula is the *continuous-time
aliasing theorem*, which lies at the foundation of elementary
*sampling theory* [264, Appendix G]. If
denotes a
continuous-time signal, its sampled version
,
, is
associated with the continuous-time signal

(B.60) |

where denotes the (fixed) sampling interval in seconds. The sampled signal values are thus treated mathematically as coefficients of impulses at the sampling instants. Taking the Fourier transform gives

where
denotes the sampling rate
in radians per second. Note that
is *periodic*
with period
. We see that if
is bandlimited to
less than
radians per second, *i.e.*, if
for all
, then only the
term will be
nonzero in the summation over
, and this means there is *no
aliasing*. The terms
for
are all
*aliasing terms*.

## The Uncertainty Principle

The *uncertainty principle* (for Fourier transform pairs) follows
immediately from the scaling theorem (§B.4). It may be
loosely stated as

where is some constant determined by the precise definitions of ``duration'' in the time domain and ``bandwidth'' in the frequency domain.Time Duration Frequency Bandwidth c

If duration and bandwidth are defined as the ``nonzero interval,'' then we obtain , which is not useful. This conclusion follows immediately from the definition of the Fourier transform and its inverse (§2.2).

### Duration and Bandwidth as Second Moments

More interesting definitions of duration and bandwidth are obtained
using the normalized *second moments* of the squared magnitude:

where

By the DTFT power theorem (§2.3.8), we have
. Note that writing ``
'' and
``
'' is an abuse of notation, but a convenient one.
These duration/bandwidth definitions are routinely used in physics,
*e.g.*, in connection with the *Heisenberg uncertainty principle* [59].Under these definitions, we have the following theorem
[202, p. 273-274]:

**Theorem: **If
as
, then

with equality if and only if

(B.63) |

That is, only the

*Gaussian function*(also known as the ``bell curve'' or ``normal curve'') achieves the lower bound on the time-bandwidth product.

*Proof: *Without loss of generality, we may take consider
to be real
and normalized to have unit
norm (
). From the
Schwarz inequality [264],^{B.2}

The left-hand side can be evaluated using integration by parts:

(B.65) |

where we used the assumption that as .

The second term on the right-hand side of (B.65) can be evaluated using the power theorem and differentiation theorem (§B.2):

(B.66) |

Substituting these evaluations into (B.65) gives

(B.67) |

Taking the square root of both sides gives the uncertainty relation sought.

If equality holds in the uncertainty relation (B.63), then (B.65) implies

(B.68) |

for some constant , implying for some constants and .

### Time-Limited Signals

If for , then

(B.69) |

where is as defined above in (B.62).

*Proof: *See [202, pp. 274-5].

### Time-Bandwidth Products Unbounded Above

We have considered two lower bounds for the time-bandwidth product
based on two different definitions of duration in time. In the
opposite direction, there is *no upper bound* on time-bandwidth
product. To see this, imagine filtering an arbitrary signal with an
*allpass filter*.^{B.3} The allpass filter cannot affect
bandwidth
, but the duration
can be arbitrarily extended by
successive applications of the allpass filter.

## Relation of Smoothness to Roll-Off Rate

In §3.1.1, we found that the side lobes of
the rectangular-window transform ``roll off'' as
. In this
section we show that this roll-off rate is due to the *amplitude
discontinuity* at the edges of the window. We also show that, more
generally, a discontinuity in the
th derivative corresponds to a
roll-off rate of
.

The Fourier transform of an impulse is simply

(B.70) |

by the sifting property of the impulse under integration. This shows that an impulse consists of Fourier components at all frequencies in equal amounts. The roll-off rate is therefore

*zero*in the Fourier transform of an impulse.

By the *differentiation theorem* for Fourier transforms
(§B.2), if
, then

(B.71) |

where . Consequently, the integral of transforms to :

(B.72) |

The integral of the impulse is the

*unit step function*:

(B.73) |

Therefore,

^{B.4}

(B.74) |

Thus, the unit step function has a roll-off rate of dB per octave, just like the rectangular window. In fact, the rectangular window can be synthesized as the superposition of two step functions:

(B.75) |

Integrating the unit step function gives a

*linear ramp function*:

(B.76) |

Applying the integration theorem again yields

(B.77) |

Thus, the linear ramp has a roll-off rate of dB per octave. Continuing in this way, we obtain the following Fourier pairs:

Now consider the Taylor series expansion of the function at :

(B.78) |

The derivatives up to order are all zero at . The th derivative, however, has a discontinuous jump at . Since this is the only ``wideband event'' in the signal, we may conclude that a discontinuity in the th derivative corresponds to a roll-off rate of . The following theorem generalizes this result to a wider class of functions which, for our purposes, will be spectrum analysis window functions (before sampling):

**Theorem: **(*Riemann Lemma*):
If the derivatives up to order
of the function
exist and
are of bounded variation (defined below), then its Fourier Transform
is asymptotically of order^{B.5}
, *i.e.*,

(B.79) |

**Proof:**Following [202, p. 95], let be any real function of bounded variation on the interval of the real line, and let

(B.80) |

denote its decomposition into a nondecreasing part and nonincreasing part .

^{B.6}Then there exists such that

Since

(B.82) |

we conclude

(B.83) |

where , which is finite since is of bounded variation. Note that the conclusion holds also when . Analogous conclusions follow for

**im**,

**re**, and

**im**, leading to the result

(B.84) |

If in addition the derivative is bounded on , then the above gives that its transform is asymptotically of order , so that . Repeating this argument, if the first derivatives exist and are of bounded variation on , we have .

Since spectrum-analysis windows
are often obtained by
*sampling* continuous time-limited functions
, we
normally see these asymptotic roll-off rates in *aliased*
form, *e.g.*,

(B.85) |

where denotes the sampling rate in radians per second. This aliasing normally causes the roll-off rate to ``slow down'' near half the sampling rate, as shown in Fig.3.6 for the rectangular window transform. Every window transform must be continuous at (for finite windows), so the roll-off envelope must reach a slope of zero there.

In summary, we have the following Fourier rule-of-thumb:

(B.86) |

This is also dB per

*decade*.

To apply this result to estimating FFT window roll-off rate
(as in Chapter 3), we normally only need to look at the window's
*endpoints*. The interior of the window is usually
differentiable of all orders. For discrete-time windows, the roll-off
rate ``slows down'' at high frequencies due to aliasing.

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Beginning Statistical Signal Processing

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Notation