Chapter 12: Properties of The Fourier Transform

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Chapter 12: Properties of The Fourier Transform
12.A Introduction.

     The power of the Fourier transform derives principally from the many
theorems describing the properties of the transformation operation which
provide insight into the nature of physical systems. Most of these theorems have
been derived within the context of communications engineering to answer
questions framed like "if a time signal is manipulated in such-and-such a way,
what happens to its Fourier spectrum?" As a result, a way of thinking about the
transformation operation has developed in which a Fourier transform pair
 y(t)  Y ( f ) is like the two sides of a coin, with the original time or space signal
on one side and its frequency spectrum on the other. The two halves of a Fourier
transform pair are thus complementary views of the same signal and so it makes
sense that if some operation is performed on one half of the pair, then some
equivalent operation is necessarily performed on the other half.

    Many of the concepts underlying the theorems and properties described
below were introduced in Chapter 6 in the context of Fourier series. For the most
part, these theorems can be extended into the domain of the Fourier transform
simply by examining the limit as the length of the observation interval for the
signal grows without bound. Consequently, it will be sufficient here simply to
list the results. Rigorous proofs of these theorems may be found in most
standard textbooks (e.g. Bracewell).


12.B Theorems

Linearity

Scaling a function scales it's transform pair. Adding two functions corresponds
to adding the two frequency spectra.

If   h( x )  H ( f )               then    ah( x )  aH ( f )                    [12.1]

     h( x )  H ( f )
If                                 then    h( x ) + g( x )  H ( f ) + G( f )    [12.2]
     g( x )  G ( f )

Scaling

Multiplication of the scale of the time/space reference frame changes by the
factor s inversely scales the frequency axis of the spectrum of the function.

If   h( x )  H ( f )               then    h( x / s )  s H( f  s)               [12.3]

                                   and     s h( x  s)  H( f / s )               [12.4]
Chapter 12: Properties of The Fourier Transform                                             Page 109


A simple, but useful, implication of this theorem is that if h( x )  H ( f ) then
h( -x )  H( - f ) . In words, flipping the time function about the origin
corresponds to flipping its spectrum about the origin.

Notice that this theorem differs from the corresponding theorem for discrete
spectra (Fig. 6.3) in that the ordinate scales inversely with the abscissa. This is
because the Fourier transform produces a spectral density function rather than a
spectral amplitude function, and therefore is sensitive to the scale of the frequency
axis.

Time/Space Shift

Displacement in time or space induces a phase shift proportional to frequency
and to the amount of displacement. This occurs because a given displacement
represents more cycles of phase shift for a high-frequency signal than for a low-
frequency signal.

     h( x )  H ( f )                               h( x - x 0 )  e- i2      fx 0
If                                       then                                      H( f )      [12.5]

Frequency Shift

Displacement in frequency multiplies the time/space function by a unit phasor
which has angle proportional to time/space and to the amount of displacement.

     h( x )  H ( f )                               h( x )ei2            H( f - f0 )
                                                               xf0
If                                       then                                                  [12.6]

Modulation

Multiplication of a time/space function by a cosine wave splits the frequency
spectrum of the function. Half of the spectrum shifts left and half shifts right.
This is simply a variant of the shift theorem which makes use of Euler's
relationship cos( x ) = (e ix + e -ix )/ 2

                                                   h( x )ei2            H( f - f0 )
                                                               xf0

if   h( x )  H ( f )                     then
                                                   h( x)e -i 2          H( f + f0 )
                                                                 xf0



and therefore by the linearity theorem it follows that

                          H ( f - f0 ) + H ( f + f 0 )
     h( x )cos(2 xf0 ) 
                                       2

                          H ( f - f 0) + H ( f + f0 )
     h( x )sin(2 xf0 )                                                                         [12.7]
                                      2i

The modulation theorem is the basis of transmission of amplitude-modulated
radio broadcasts. When a low frequency audio signal is multiplied by a radio-
frequency carrier wave, the spectrum of the audio message is shifted to the radio
portion of the electromagnetic spectrum for transmission by an antenna.
Chapter 12: Properties of The Fourier Transform                                      Page 110


Similarly, a method for recovery of the audio signal called super-heterodyne
demodulation involves multiplying the received signal by a sinusoid with the
same frequency as the carrier, thereby demodulating the audio component.

Differentiation

Differentiation of a function induces a 90 phase shift in the spectrum and scales
the magnitude of the spectrum in proportion to frequency.                Repeated
differentiation leads to the general result:

                                             d n h( x)
If   h( x )  H ( f )                 then          n    (i 2 f )n H ( f )                [12.8]
                                                dx

This theorem explains why differentiation of a signal has the reputation for being
a noisy operation. Even if the signal is band-limited, noise will introduce high
frequency signals which are greatly amplified by differentiation.

Integration

Integration of a function induces a -90 phase shift in the spectrum and scales the
magnitude of the spectrum inversely with frequency.
                                                x
If   h( x )  H ( f )                 then       -
                                                    h(u ) du  H ( f )/(i 2 f ) + constant [12.9]

From this theorem we see that integration is analagous to a low-pass filter which
blurs the signal.

Transform of a transform

We normally think of using the inverse Fourier transform to move from the
frequency spectrum back to the time/space function. However, if instead the
spectrum is subjected to the forward Fourier transform, the result is a time/space
function which has been flipped about the y-axis. This gives some appreciation
for why the kernels of the two transforms are complex conjugates of each other:
the change in sign in the reverse transform flips the function about the y-axis a
second time so that the result matches the original function.

     h(t )    H( f )                         H( f )    h(-t )
           F                                        F
If                                   then                                               [12.10]

One practical implication of this theorem is a 2-for-1 bonus: every transform pair
brings with it a second transform pair at no extra cost.

If       h(t )  H ( f )              then    H(t )  h( - f )                            [12.11]

For example, rect( t )  sinc( f ) implies sinc(t)  rect(- f ) .

This theorem highlights the fact that the Fourier transform operation is
fundamentally a mathematical relation that can be completely divorced from the
physical notions of time and frequency. It is simply a method for transforming a
Chapter 12: Properties of The Fourier Transform                                                    Page 111


function of one variable into a function of another variable. So, for example, in
probability theory the Fourier transform is used to convert a probability density
function into a moment-generating function, neither of which bear the slightest
resemblance to the time or frequency domains.

Central ordinate

By analogy with the mean Fourier coefficient a0 , the central ordinate value H(0)
(analog of a0 in discrete spectra) represents the total area under the function h(x).
                                                                                      

                                                               - h(u) e du =       
                                                                       -i 0
If   h( x )  H ( f )                      then   H(0) =                                  h(u) du
                                                                                    -


                                                               
                                                 h(0) = - H (u) ei 0du
                                                               
For the inverse transform,                               = - H(u) du                                     [12.12]
                                                                                         
                                                         =   - Re[H (u )] du + i - Im[ H(u )] du
Note that for a real-valued function h(t) the imaginary portion of the spectru will
have odd symmetry, so the area under the real part of the spectrum is all that
needs to be computed to find h(0).

For example, in optics the line-spread function (LSF) and the optical transfer
function (OTF) are Fourier transform pairs. Therefore, according to the central-
ordinat theorem, the central point of the LSF is equal to the area under the OTF.
In two dimensions, the transform relationship exists between the point-spread
function (PSF) and the OTF. In such 2D cases, the integral must be taken over an
area, in which case the result is interpreted as the volume under the 2D surface.

Equivalent width

A corollary the central ordinate theorm is
        
     
                                                    
            h(u ) du = H(0)
If
        -
                                          then
                                                    -
                                                        h(u ) du
                                                                   =
                                                                           H (0)
                                                                                                     [12.13]
                                                                        

                                                                       - H (u) du
                           
                        
               h(0) =          H (u) du                 h(0)
                           -


The ratio on the left side of this last expression is called the "equivalent width" of
the given function h because it represents the width of a rectangle with the same
central ordinate and the same area as h. Likewise, the ratio on the right is the
inverse of the equivalent width of H. Thus we conclude that the equivalent
width of a function in one domain is the inverse of the equivalent width in the
other domain as illustrated in Fig. 12.1. For example, as a pulse in the time
domain gets shorter, its frequency spectrum gets longer. This theorem quantifies
that relationship for one particular measure of width.
Chapter 12: Properties of The Fourier Transform                               Page 112


                   Fig. 12.1 Equivalent Width Theorem
        Space/time Domain                     Frequency Domain



                  w                                                1
                                                                   w


Convolution

The convolution operation (denoted by an asterisk) is a way of combining two
functions to produce a new function. By definition,
                                                      
     p = hg                   means          p( x ) = - g(u)h( x - u ) du       [12.14]

Convolution will be described in detail in section 12C. Here it is sufficient to
state the convolution theorem:

     h( x )  H ( f )                         h( x )  g( x)  H ( f )  G( f )
If                                    then                                      [12.15]
     g( x )  G ( f )                         h( x )  g( x)  H( f )  G( f )

In words, this theorem says that if two functions are multiplied in one domain,
then their Fourier transforms are convolved in the other domain. Unlike the
cross-correlation operation described next, convolution obeys the commutiative,
associative, and distributive laws of algebra. That is,

     commutative law                  hg = gh
     associative law                  f  (g  h) = ( f  g)  h                    [12.16]
     distributive law                 f  ( g + h) = f  g + f  h

Derivative of a convolution

Combining the derivative theorem with the convolution theorm leads to the
conclusion

                                             dh df        dg
If   h( x ) = f ( x)  g(x )           then     =   g = f                        [12.17]
                                             dx dx        dx

In words, this theorem states that the derivative of a convolution is equal to the
convolution of either of the functions with the derivative of the other.

Cross-correlation

The cross-correlation operation (denoted by a pentagram) is a way of combining
two functions to produce a new function that is similar to convolution. By
definition,
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