9Fourier TransformProperties

The Fourier transform is a major cornerstone in the analysis and representa-tion of signals and linear, time-invariant systems, and its elegance and impor-tance cannot be overemphasized. Much of its usefulness stems directly fromthe properties of the Fourier transform, which we discuss for the continuous-time case in this lecture. Many of the Fourier transform properties might atfirst appear to be simple (or perhaps not so simple) mathematical manipula-tions of the Fourier transform analysis and synthesis equations. However, inthis and later lectures, as we discuss issues such as filtering, modulation, andsampling, it should become increasingly clear that these properties all haveimportant interpretations and meaning in the context of signals and signalprocessing.

The first property that we introduce in this lecture is the symmetry prop-erty, specifically the fact that for time functions that are real-valued, the Four-ier transform is conjugate symmetric, i.e., X( - o) = X*(w). From this it fol-lows that the real part and the magnitude of the Fourier transform of real-valued time functions are even functions of frequency and that the imaginarypart and phase are odd functions of frequency. Because of this property ofcorjugate symmetry, in displaying or specifying the Fourier transform of areal-valued time function it is necessary to display the transform only forpositive values of w.

A second important property is that of time and frequency scaling, spe-cifically that a linear expansion (or contraction) of the time axis in the timedomain has the effect in the frequency domain of a linear contraction (expan-sion). In other words, linear scaling in time is reflected in an inverse scaling infrequency. As we discuss and demonstrate in the lecture, we are all likely tobe somewhat familiar with this property from the shift in frequencies that oc-curs when we slow down or speed up a tape recording. More generally, this isone aspect of a broader set of issues relating to important trade-offs betweenthe time domain and frequency domain. As we will see in later lectures, forexample, it is often desirable to design signals that are both narrow in timeand narrow in frequency. The relationship between time and frequency scal-ing is one indication that these are competing requirements; i.e., attempting

Signals and Systems9-2

to make a signal narrower in time will typically have the effect of broadeningits Fourier transform.

Duality between the time and frequency domains is another importantproperty of Fourier transforms. This property relates to the fact that the anal-ysis equation and synthesis equation look almost identical except for a factorof 1/2 7r and the difference of a minus sign in the exponential in the integral. Asa consequence, if we know the Fourier transform of a specified time function,then we also know the Fourier transform of a signal whose functional form isthe same as the form of this Fourier transform. Said another way, the Fouriertransform of the Fourier transform is proportional to the original signal re-versed in time. One consequence of this is that whenever we evaluate onetransform pair we have another one for free. As another consequence, if wehave an effective and efficient algorithm or procedure for implementing orcalculating the Fourier transform of a signal, then exactly the same procedurewith only minor modification can be used to implement the inverse Fouriertransform. This is in fact very heavily exploited in discrete-time signal analy-sis and processing, where explicit computation of the Fourier transform andits inverse play an important role.

There are many other important properties of the Fourier transform,such as Parseval's relation, the time-shifting property, and the effects on theFourier transform of differentiation and integration in the time domain. Thetime-shifting property identifies the fact that a linear displacement in timecorresponds to a linear phase factor in the frequency domain. This becomesuseful and important when we discuss filtering and the effects of the phasecharacteristics of a filter in the time domain. The differentiation property forFourier transforms is very useful, as we see in this lecture, for analyzing sys-tems represented by linear constant-coefficient differential equations. Also,we should recognize from the differentiation property that differentiating inthe time domain has the effect of emphasizing high frequencies in the Fouriertransform. We recall in the discussion of the Fourier series that higher fre-quencies tend to be associated with abrupt changes (for example, the step dis-continuity in the square wave). In the time domain we recognize that differen-tiation will emphasize these abrupt changes, and the differentiation propertystates that, consistent with this result, the high frequencies are amplified inrelation to the low frequencies.

Two major properties that form the basis for a wide array of signal pro-cessing systems are the convolution and modulation properties. According tothe convolution property, the Fourier transform maps convolution to multi-plication; that is, the Fourier transform of the convolution of two time func-tions is the product of their corresponding Fourier transforms. For the analy-sis of linear, time-invariant systems, this is particularly useful becausethrough the use of the Fourier transform we can map the sometimes difficultproblem of evaluating a convolution to a simpler algebraic operation, namelymultiplication. Furthermore, the convolution property highlights the fact thatby decomposing a signal into a linear combination of complex exponentials,which the Fourier transform does, we can interpret the effect of a linear, time-invariant system as simply scaling the (complex) amplitudes of each of theseexponentials by a scale factor that is characteristic of the system. This "spec-trum" of scale factors which the system applies is in fact the Fourier trans-form of the system impulse response. This is the underlying basis for the con-cept and implementation of filtering.

The final property that we present in this lecture is the modulation prop-erty, which is the dual of the convolution property. According to the modula-tion property, the Fourier transform of the product of two time functions is

Fourier Transform Properties9-3

proportional to the convolution of their Fourier transforms. As we will see ina later lecture, this simple property provides the basis for the understandingand interpretation of amplitude modulation which is widely used in communi-cation systems. Amplitude modulation also provides the basis for sampling,which is the major bridge between continuous-time and discrete-time signalprocessing and the foundation for many modern signal processing systemsusing digital and other discrete-time technologies.

We will spend several lectures exploring further the ideas of filtering,modulation, and sampling. Before doing so, however, we will first develop inLectures 10 and 11 the ideas of Fourier series and the Fourier transform forthe discrete-time case so that when we discuss filtering, modulation, and sam-pling we can blend ideas and issues for both classes of signals and systems.

Suggested ReadingSection 4.6, Properties of the Continuous-Time Fourier Transform, pages

202-212

Section 4.7, The Convolution Property, pages 212-219Section 6.0, Introduction, pages 397-401Section 4.8, The Modulation Property, pages 219-222Section 4.9, Tables of Fourier Properties and of Basic Fourier Transform and

Fourier Series Pairs, pages 223-225Section 4.10, The Polar Representation of Continuous-Time Fourier Trans-

forms, pages 226-232Section 4.11.1, Calculation of Frequency and Impulse Responses for LTI Sys-

tems Characterized by Differential Equations, pages 232-235

Signals and Systems

TRANSPARENCY9.1Analysis and synthesisequations for thecontinuous-timeFourier transform.

CONTINUOUS - TIME FOURIER TRANSFORM

=1X(t)

+00

X(G)= f

+00

00X(co) e jot dco

x(t) e~jot dt

x(t) +->.

synthesis

analysis

X(W)

X(w) = Re IX(w) + j

= IX(eo)ej x("

Im (j)[

TRANSPARENCY9.2Symmetry propertiesof the Fouriertransform.

PROPERTIES OF THE FOURIER TRANSFORM

X(t) X(j)

Symmetry:

x(t) real

Re X(o)

=> X(-w) = X*()

= Re X(-o)even

IX(co)I = IX(-w)I

Im X(o)

'4X(o)

= -Im X(-4)odd

Fourier Transform Properties

9-5

Example 4.7: eat u(t) a > oa+jcw

TRANSPARENCY

IxMe) 9.31/a The Fourier transform

1/a/2 for an exponential- -- time function

illustrating theproperty that theFourier transformmagnitude is even and

1/a -a a the phase is odd.

7r/4

-7/4

IT/2

Time and frequency scaling:TRANSPARENCY

1 \9.4x(at) -+ X The property of time

and frequency scaling

Example: for the Fouriertransform.

&>tut)~1 _1 1e- at Ut L -I- -a+jo a 1+ j (J

1/a

x(t)

et C

Signals and Systems

DEMONSTRATION9.1Time and frequencyscaling. A glockenspielnote is recorded andthen replayed at twiceand half speed.

TRANSPARENCY9.5The property ofduality.

I

Fourier Transform Properties

Example 4.11:

2sin wt2 2rt

Example 4.10:

X(W)

w w

X((o)

2co

TRANSPARENCY9.6Illustration of theduality property.

+00

f Ix(t)|2dt00

if I'(t)I12dt0O

127r

+00)

f-00

+00

k=- 00

IX(c) |2do

lak12

TRANSPARENCY9.7Parseval's relation forthe Fourier transformand the Fourier series.

Parseval's relation:

Signals and Systems

9-8

TRANSPARENCY9.8

Some additionalproperties of theFourier transform.

Time shifting:

x(t-t ) .- e-jwto X(o)

Differentiation:

dx(t)dt jw X(W)

Integration:

f tf00 - X(W) + 7r X(O) b(w)Jco

Linearity:

ax (t) + bx 2 (t) -. aX1 (w) + bX 2 (w)

TRANSPARENCY9.9Transparencies 9.9and 9.10 illustrate theconvolution propertyand its interpretationfor LTI systems. Thistransparency indicatesthe response to animpulse.

x(T)dr 4-+

Fourier Transform Properties

CONVOLUTION PROPERTY

h(t) * x(t) H (w) X( )

x(t) h(t) h(t) * x(t)

X(W) H(w) H(w) X(W)

e*Jcot H(w o )

H(w) = frequency response

TRANSPARENCY9.10Response to acomplex exponential.

FILTERING

Ideal lowpass filter:

-IDifferentiator:

dx(t)y(t) =t => H(co) = jco

TRANSPARENCY9.11Filtering as anillustration of theinterpretation of theconvolution propertyfor LTI systems.Co Co

IH(M)I

Oppo

Signals and Systems9-10

DEMONSTRATION9.2Lowpass filtering of animage.

DEMONSTRATION9.3Effect ofdifferentiating animage.

Fourier Transform Properties9-11

MODULATION PROPERTY

s(t) p(t)

s(t) p(t)

Convolution: s(t) * p(t)

1-- [S(o) * P(w)]

27r

-- [S(w) * P(co)]27r

S(co) P(W)

TRANSPARENCY9.12The modulationproperty for theFourier transform.

_+)+ Gt (t) =At

(7): (V

j. J)j &o ,e

-t

%~, %~'

I IY~wj~ jQ4I

- I I=,ji342. Jh~+I

t ~

Modulation:

MARKERBOARD9.1

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