BYST SigSys - WS2003: Fourier Rep. 120 CPE200 Signals and Systems Chapter 3: Fourier Representations for Signals (Part I)

BYSTBYSTSigSys - WS2003: Fourier Rep.SigSys - WS2003: Fourier Rep. 1

CPE200 Signals CPE200 Signals and Systemsand Systems

Chapter 3: Fourier Representations fo

r Signals(Part I)


Introduction3.1

A common engineeringengineering analysis techniqueanalysis technique is the partitioning of complex problems into a linear combination of the simpler ones. The simpler problems are then solved, and the total solution becomes the sum of the simpler solutions. Two requirements must be satisfied for a solution as described earlier to be both valid and useful. First, the problem must be linearlinear, such that the solution for the complex problem is equal to the sum of the solutions considering only one simpler problem at a time. Next, the complex problem must be able to express as express as a number of simpler problemsa number of simpler problems.

Analysis techniques in signal and system theory follow the same manner mentioned above. A linear system will allow us to


determine the response of each individual input when the input is a combination of more than one signals. The only required procedure is how to decompose the complicated signal into a combination of simpler signals.

Any signals can also be represented as a weighted superposition of complex sinusoids. From Chapter 2, we discovered that any signals can be represented as linear combinations of shifted impulses. When such a signal is applied to a linear time-invariant system, then the system response is based on the convolution sum. However, there is an alternative representation for signals and systems which will be discussed in this chapter and the following two chapters. The alternative representation for signals is known as the c-t and d-t Fourier series Fourier series and transformtransform.


The Fourier seriesFourier series is named after the French physicist Jean Baptiste Fourier (1768-1830), who was the first one to propose that perperiodiciodic waveforms could be represented by a sum of sinusoids (or complex exponential signals). When the signals are aperiodicaperiodic and have finite energy finite energy the Fourier series is known as the Fourier transform Fourier transform. A signal with such decomposition is said to be represented in the frequency domainfrequency domain.

By representing signals in terms of sinusoids, we will obtain an alternative expression for the input-output behavior of an LTI system which is a weighted sum of the system response to each complex sinusoidal. This type of representation not only leads to a useful expression for the system response but also provides a very


perceptive characterization of signals and systems.

In this chapter, we focus on the Fourier representations of signals and their properties. We will begin our discussion on studying the response of LTI systems to complex sinusoids. Then, we will consider the development of a representation of signals as linear combinations of complex sinusoids. Properties of c-t and d-t Fourier representations will also be discussed. Finally, the basic concept of filtering will be introduced.

Complex Sinusoids and3.2LTI Systems

Sinusoidal input signals are often used to characterized the response of a system. In Chapter 2, we discussed how to


characterize an LTI system using the impulse and unit step signals. We discovered that when the impulse response is known, we can determine the response of an LTI system to any arbitrary input by a weighted sum (integral) of time-shifted impulse responses. In this section, we will examine the relationship between the impulse response and the steady-state response of an LTI system to a complex sinusoidal input.

Let consider the d-t complex sinusoidalcomplex sinusoidal input expressed as x[n] = e x[n] = e jjnn. Hence the response y[n] is given by

k

]kn[x]k[h]n[y


k

)kn(je]k[h

k

kjnj e]k[he (3.1)

Let H(eH(ejj)) be the “frequency responsefrequency response” of an LTI system and define as

k

kjj e]k[h)e(H (3.2)

Substitute H(ej) in Eq. 3.1, we obtain

njj e)e(H]n[y (3.3)

Frequency Frequency ResponseResponse

Input x[n]Input x[n]


Eq. 3.3 states that the response of an LTI system to a complex sinusoid is also , e e jjnn, a complex sinusoid of the same frequencythe same frequency as the input multiplied by the frequencyresponse H(ej). Fig. 3.1 illustrates the response of an LTI system to a complex sinusoidal input.

h[n]e jn H(ej) e jn

Figure 3.1 A complex sinusoidal d-t input to an LTI systemresults in a complex sinusoidal output of the samefrequency multiplied by the frequency responseof the system.

The analogous relationship holds in the c-t case as shown in Fig. 3.2.


h(t)e jt H(j) e jt

Figure 3.2 A complex sinusoidal c-t input to an LTI systemresults in a complex sinusoidal output of the samefrequency multiplied by the frequency responseof the system.

The complex sinusoidal input in Eq. 3.3 is called an eigenfunctioneigenfunction of the system since it produces an output that differs from the input by a constant multiplicative factor H(ej). This multiplicative factor is called an eeigenvalueigenvalue of the system.

We can notice that the frequency response H(ej) is a complex valuea complex value. That is, if we represent H(ej) in complex form as shown in Eq. 3.4, then we may express H(ej) in polar form as shown in Eq. 3.5.

H(ej) = HR(ej) + HI(ej) (3.4)


)]}e(H{arg[jjj j

e|)e(H|)e(H (3.5)

Magnitude Magnitude ResponseResponse

Phase ResPhase Responseponse

where

HR(ej) = the real part of H(ej),

HI(ej) = the imaginary part of H(ej),

|H(ej)| = the magnitude responsethe magnitude response

,)e(H)e(H j2I

j2R

(the magnitude of H(ej))

arg[H(ej)] = the phase responsethe phase response(the phase of H(ej))

.)e(H

)e(Htan

jR

jI1


For the analysis LTI system with the complicated signal, let consider x[n] represented as weighted sum of N complex sinusoids:

(3.6)

njN

nj2

nj1

N21 ea...eaea]n[x

N

1k

njk

kea

If is an eigenfunction of the system with eigenvalue H(ej), then each term in the input as indicated by Eq. 3.6, produces an output term .

nj ke

njjk

ke)e(Ha

Therefore, the output to x[n] defined by Eq. 3.6 can be evaluated as follows:

N

1k

njk

j kk ea)e(H]n[y (3.7)


That is, when the input x[n] is expressed as a sum of eigenfunctions, the output is also a weighted sum of N complex sinusoids, with the weights ak modified by the frequency response H(ej). This property describes the signal behavior as a function of frequency which will be discussed later in this chapter.

3.2.1 Response to an ExponentiallyDamped Sinusoidal

Previously, we only considered a complex exponential signal of the form e e st st with s s = j= j. Such signal is a pure imaginarypure imaginary complex exponential signal. In this section, we will observe the response of an LTI system to a generalization of the complex exponential signal where s = s = +j+j.


Let consider the c-t exponentially damped exponentially damped sinusoidalsinusoidal input expressed as x(t) = ex(t) = estst. Hence, from Eq. 2.9, the response of the LTI system can be evaluated as follows:

s(t )h( )e d

st se h( )e d

(3.8)

y(t) h( )x(t )d

sty(t) H(s) e (3.9)

TransferTransferFunctionFunction

Input x(t)Input x(t)

Eq. 3.8 can be rewritten in a simply form as:


sH(s) h( )e d

(3.10)

where H(s)H(s) is called “transfer functiontransfer function” of an LTI system which is related to the system impulse response, h(t), by

Eq. 3.9 states that the response of an LTI system to an exponentially damped sinusoid is also , e e stst, an exponentially damped sinusoid of the same complex frequencythe same complex frequency as the input multiplied by the transfer function H(s). That is, the constant H(s)H(s) for a specific value of ss is then the eigenvalueeigenvalue associated with the eigenfunctioneigenfunction est.

Similarly, the analogous relationship holds in the d-t case when the input sequence of a d-t LTI system is of the form: x[n]=zx[n]=znn. The


response of the system can be determined from the following equation:

ny[n] H(z) z (3.11)

SystemSystemFunctionFunction

Input x(t)Input x(t)

where H(z)H(z) is called “system functionsystem function” of a d-t LTI system which is related to the system unit-sample response, h[n], by

k

k

H(z) h[k]z

(3.12)

Similarly, the constant H(z)H(z) for a specific value of zz is then the eigenvalueeigenvalue associated with the eigenfunctioneigenfunction zn.


3.2.2 Response to a Sinusoidal Signal

Sinusoidal signals are the basic building blocks for generating more complicated signals. That is, knowing the response of LTI systems to a sinusoidal signal is so important. So far, we discovered that the response of an LTI system to the complex exponential signal eejjnn (eejjtt) or eestst or zznn is also the complex exponential signal multiplied by H(eH(ejj)) or H(s)H(s) or H(z)H(z), respectively. Thus, if the input to an LTI system is a real-valued sinusoidal signal, the response of the system should result in the same manner since, by Euler's formula, any sinusoidal signals can be represented as a combination of complex exponential signals.

Let x[n] be a sinusoidal signal;


Using Euler’s relation, the real-valued sinusoidal signal can be rewritten as:

0 0j( t ) j( t )A Ax[n] e e

2 2 (3.14)

From Eq. 3.14, x[n] is now represented as a linear combination of two complex exponential signals which are complex conjugate to each other. Hence, from Eq. 3.7, the response of an LTI system to x[n] defined by Eq. 3.14 can be evaluated as follows:

(3.15)

njjj 00 e)e(He2

A]n[y

njjj 00 e)e(He2

A

)ncos(A]n[x 0 (3.13)


From Eq. 3.15, the second term is the complex conjugate of the first term. Therefore, only the real part of either term will be remained. That is,

0j nj jAy[n] 2 e H(e )e e

2

ncos[|)e(H|A]n[y 0j 0

)}]e(Harg{ 0j

(3.16)

From Eq. 3.16, the response of an LTI system to the sinusoidal signal is another sinusoid of the same frequencysame frequency but with different different magnitudemagnitude and phase phase. The magnitude of the input sinusoid is modified by the magnitude response, |H(e|H(ejj)|)| and the


phase of the input sinusoid is modified by the phase response, arg{H(earg{H(ejj)})}.

The frequency response characterize the steady-state response of the system to sinusoidal inputs as a function of the sinusoid’s frequency. It is said to be a steady-state response since the input sinusoid is assumed to exist for all time and thus the system is in an equilibrium condition.

Fourier Series Representation3.3of C-T Periodic Signals

Fourier series (FS) are series of sinusoidal or complex exponential signals. It was first introduced by Jean Baptiste Joseph, Baron de Fourier who discovered that a complicate periodic signal (with a pattern


that repeats itself) consists of the sum of many simple waves. He suggested that most signals can be represented by a sum of a sum of sinusoidal signal with different frequenciessinusoidal signal with different frequencies. Hence, he has purposed a method to decompose a given periodic signal into a linear combination of sinusoidal signals having different frequencies. The resulting sum is called the Fourier seriesFourier series or the Fourier expansionFourier expansion.

A fundamental concept in the study of Fourier series is the spectrum spectrum which is the notation of the frequency contentfrequency content of a signal. For a large class of signals, the frequency content can be evaluated by decomposing the signal into frequency components given

3.3.1 Representation of Signals inTerms of Frequency Components


by sinusoidal signals. In this section, the most general and powerful method for generating new signals from sinusoidal signals will be introduced. This method will create any signals by adding together a constant and N sinusoids, each with a different frequency, amplitude and phase as shown in Eq. 3.17. Mathematically, this signal may be represented by the equation

N

0 k k kk 1

x(t) A A cos(2 F t )

(3.17)

Using Euler’s relation, Eq. 3.17 can be rewritten as:

k k

Nj j2 F tk

0k 1

Ax(t) A e e

2

(3.18)k kj j2 F tkAe e

2


Let XXkk represent the phasorphasor of the individual sinusoidal components. That is,

kjk kX A e (3.19)

Substitute Eq. 3.19 into Eq. 3.18, the alternative form of x(t) can be represented as follows:

k

Nj2 F tk

0k 1

Xx(t) X e

2

(3.20)k

*j2 F tkX

e2

where X0 = A0.

We see then that each sinusoideach sinusoid in the sum decomposes into two rotating phasorstwo rotating phasors, one with positive frequency, FFkk, and the other with negative frequency, -F-Fkk. This form


follows the fact that the real part of a complex number is equal to one-half the sum of that number and its complex conjugate.

Eq. 3.20 indicates the frequency domain frequency domain representationrepresentation of the signal x(t) in terms of a pair (0.5X(0.5Xkk, F, Fkk)) which is usually called the

(frequency) spectrum(frequency) spectrum. That is, the spectrum is just the set of pairs:

The spectrum of signal x(t) simply provides the information required to synthesize it. The graphical plot of the spectrum can be illustrated in Fig. 3.3.

{(X0, 0), (0.5X1, F1), (0.5X*1, -F1),

(0.5X2, F2), (0.5X*2, -F2), … }.


F (Hz)

X0

0 F1 F2 F3-F1-F2-F3

......

0.5X10.5X*1

0.5X2

0.5X3

0.5X*2

0.5X*3

Figure 3.3 Spectrum of the signal x(t) represented by Eq. 3.20

A general procedure for computing and plotting the spectrum of any signal is called Fourier analysisFourier analysis. This procedure involves simply decomposing a signal x(t) into the complex exponential form as shown in Eq. 3.20, and selecting off the amplitude, phase, and frequency of each of its rotating phasor components. The mathematical tools for performing this analysis are called "Fourier Fourier seriesseries" and "Fourier transformsFourier transforms".


3.3.2 Linear Combinations ofHarmonically Related ComplexExponential

We have seen how a signal can be created by a sum of a constant and N sinusoids having difference frequency, amplitude and phase. An interesting situation occurs when a signal x(t) is a periodic signal defined as follows:

x(t) = x(t+T) for all t

When TT is the smallest positive, nonzero value, it is called the "fundamental periodfundamental period" and the value 00 = 2 = 2/T/T is referred to as the "fundamental angular frequencyfundamental angular frequency".

To maintain the fundamental period, T, of x(t), the frequency Fk in Eq. 3.20 must be the kth harmonic frequency of the


fundamental frequency F0 of x(t). That is, the signal x(t) will be a linear combinations of harmonically related sinusoidsharmonically related sinusoids. Hence, if we substitute FFkk in Eq. 3.20 with kFkF00 and allow the sum to be infinite (i.e., N = ∞), any periodic signal x(t) can be expressed as follows:

0jk tk

k

x(t) a e

(3.21)

2jk t

Tk

k

x(t) a e

0jk2 F t

kk

x(t) a e

or

or

Fourier SeriesFourier Series


k0 = the kth harmonic.

k =1k =1 FF00 ( (00) = the first harmonic) = the first harmonic(the fundamental(the fundamental(angular) frequency)(angular) frequency)

where

aakk = the Fourier series coefficientsthe Fourier series coefficients,

k 0, 1, 2,

The harmonic series defined by Eq. 3.21 is called the Fourier series (FS)Fourier series (FS) of the periodic signal x(t). Fourier series represents any periodic signals in the form of the weights (aakk) on a "discretediscrete" set of the frequencies that are integer (k) multiples of the integer (k) multiples of the fundamental (angular) frequencyfundamental (angular) frequency 00. Note that the Fourier series is a special case of Eq. 3.20 in that it is a linear combination of infinite (N = ∞) harmonically complex exponentials.


These FS coefficients aFS coefficients akk are often referred to as the spectral coefficientsspectral coefficients of x(t). In general, FS coefficients ak are complex coefficients which can be represented in a polar form of:

kjk ka A e

The magnitude of ak (Ak) is known as the mm

agnitude spectrumagnitude spectrum of x(t). Similarly, the phase of ak (k) is known as the phase spectrum phase spectrum of x(t).

If x(t) is realreal periodic signal, the Fourier series coefficient ak and a-k are a pair of complex conjugate with respect to each other. That is,


*k ka a

For a For a realreal periodic siperiodic signal,gnal,

(3.22)

The Fourier series coefficients ak of the complex exponentials in Eq. 3.21 are evaluated by the following equation:

0jk tk

T

1a x(t)e dt

T (3.23)

The coefficient a0 is the constant or de component of x(t) given by

0

T

1a x(t)dt,

T (3.23)

which is simply the average of x(t) over one period.


Alternative Forms of Fourier SeriesAlternative Forms of Fourier Series

The Fourier series described by Eq. 3.21 is in the exponential form. In the case of realreal periodic signalsperiodic signals, the other two alternative forms of the Fourier series can be derived using Euler’s relation.

Let x(t) be a real periodic signal having the Fourier series coefficients ak. Since a-k = a*

k, then -k = -k. For a given value of k, the sum of the two terms of the same frequency k0 in Eq. 3.21 yields

0 0jk t jk tk ka e a e

(3.24a)

0jk tk2Re{a e }

0 0k kjk t jk tj jk kA e e A e e or


0 k 0 kj(k t ) j(k t )kA [e e ]

k 0 k2A cos(k t )

By using Eq. 3.24 and rearranging the summation in Eq. 3.21, we can easily derive the combined trigonometric formcombined trigonometric form for the Fourier series of realreal periodic signals:

0 k 0 kk 1

x(t) a 2 A cos(k t )

(3.25)

(3.24b)

A third form for the Fourier series of realreal periodic signals can be obtained by writing ak in rectangular form as


Substituting Eq. 3.26 into Eq. 3.21 yields the trigonometric formtrigonometric form of the Fourier series

ak = Bk +jCk,

where BBkk and C and Ckk are real are real. Hence, with this expression for ak, Eq. 3.24a can be rewritten as:

0 0jk t jk tk ka e a e

k 0 k 02Re[B cos k t C sin k t

k 0 k 0jC cos k t jB sin k t]

k 0 k 02[B cos k t C sin k t] (3.26)


The exponential form (Eq. 3.21) and the combined trigonometric form (Eq. 3.25) of the Fourier series are probably the most useful forms in signal and system theory.

(3.27)

0 k 0k 1

x(t) a 2 [B cos k t

k 0C sin k t]

where

0

T

1a x(t)dt,

T

k 0

T

1B x(t) cos(k t)dt,

T

k 0

T

1C x(t)sin(k t)dt.

T


(3.28)

0 k 0k 1

x(t) a [b cos k t

k 0c sin k t]

where bbkk and c and ckk are real are real.

0

T

1a x(t)dt,

T

k k 0

T

2b 2B x(t) cos(k t)dt,

T

k k 0

T

2c 2C x(t)sin(k t)dt.

T

Note that the trigonometric form of the Fourier series expressed by Eq. 3.27 is normally rewritten as follows:


Fourier suggested that any periodic signal could be expressed as a sum of complex exponentials (or sinusoids). However, in fact, it is partially true in that Fourier series can be used to represent an extremely large class of periodic signals. In particular, a periodic signal x(t) has a Fourier series if it satisfies the Dirichlet Dirichlet conditionsconditions given by

Convergence of the Fourier SeriesConvergence of the Fourier Series

1. x(t) is absolutely integrable over any period; that is,

T

| x(t) | dt 2. x(t) has only a finite number of maxim

a and minima over any period.

(3.29)


3. x(t) has only a finite number of discontinuities over any period.

Fourier Series Representation3.4of D-T Periodic Signals

In this section, we consider the Discrete- Discrete- Time Fourier Series (DTFS)Time Fourier Series (DTFS) which is the Fourier series representation of discrete- time (d-t) periodic signals. Although, the principle concept of DTFS is similar to FS. There are some important differences. In particular, DTFS is a finite series while as FS is a infinite series. Being a finite series, DTFS always exists, as opposed to FS which requires some conditions for existence.

A d-t signal x[n] is periodic with period Nperiod N if


x[n] = x[n+N] for all n.

When NN is the smallest positive, nonzero value, it is called the "fundamental periodfundamental period" and the value 00 = 2 = 2/N/N is referred to as the "fundamental angular frequencyfundamental angular frequency".

0jk nk

k N

x[n] a e

(3.30)or

or

The DTFS represents an N periodic d-t signal x[n] is defined as the series of Eq. 3.30:

2jk n

Nk

k N

x[n] a e

0jk2 f n

kk N

x[n] a e


The D-T Fourier series coefficients ak can be evaluated from Eq. 3.31 as follows:

0jk nk

n N

1a x[n]e

N

(3.31)

As in continuous time, DTFS coefficients aakk in Eq. 3.31 are also referred to as the spectral coefficientsspectral coefficients of x[n]. These coefficients specify a decomposition of x[n] into a sum of N harmonically related complex exponentials. Each ak is associated with a complex sinusoid of a different frequency. Since there are only N distinct complex exponentials that are periodic with period N, DTFS representation is a finite series with N terms. Hence, if we consider more than N sequential values of k, the


*k ka a for a real x[n].

If x[n] is realreal periodic d-t signal, DTFS coefficient ak and a-k are a pair of complex conjugate with respect to each other. That is,

values ak repeat periodically with period N. That is,

ak = ak+N (3.32)

(3.33)

There are two alternative forms for DTFS of realreal periodic sequencesperiodic sequences which are similar to Eq. 3.25 and Eq. 3.27. If the polar form of

kjk ka A e


(N 1) / 2

0 k kk 1

2 knx[n] a 2 A cos

N

(3.34)

n0 N / 2x[n] (a a ( 1) ) (N 1) / 2

k kk 1

2 kn2 A cos

N

If we express ak in the Cartesian form of

is used, DTFS of real periodic x[n] can be expressed as:

ak = Bk +jCk

if N is odd or as

if N is even.


where BBkk and C and Ckk are real are real, the alternative form for DTFS of a real periodic x[n] can be expressed as:

(N 1) / 2

0 kk 1

2 knx[n] a 2 [B cos

N

(3.35)

n0 N / 2x[n] (a a ( 1) ) (N 1) / 2

kk 1

2 kn2 [B cos

N

if N is odd or as

if N is even.

k2 kn

C sin ]N

k2 kn

C sin ]N


The Continuous-Time Fourier3.5Transform (FT)

A key feature of the Fourier series representation of periodicperiodic signals is the description of such signals in terms of the frequency content given by sinusoidal components. Now we would like to extend this notation to nonpnonperiodiceriodic signals, also called aperiodicaperiodic signals. As will be seen, the frequency components of aperiodic signals are infinitesimally close iinfinitesimally close in frequencyn frequency rather than harmonically relatharmonically related frequencyed frequency as in the case of periodic signals. In other words, signals that are periodiperiodicc in timetime have discrete frequencydiscrete frequency domain representations, while aperiodic timeaperiodic time signals have continuous frequencycontinuous frequency domain representations. Therefore, the representation in terms of a linear combination takes the form of an


integral instead of a sum and such representation will be called the Fourier transformFourier transform.

The extension of the Fourier series to aperiodic signals is based on the idea of extending the period to infinity. Let xT(t) denote the pulse train with period T as shown in Fig. 3.4.

1

T1

1 | t | Tx (t) T

0 T | t |2

T2

TT1T2

--T1-T

......

xT(t)

Figure 3.4 A continuous-time periodic pulse train.

That is,

(3.36)


To demonstrate the change in frequency component of xT(t) as T → ∞, the Fourier series coefficients ak for this pulse train will be evaluated. Since

,dte)t(xT

1a

T

tjkk

0

then, for k = 0 gives

,T

T2dt)t(x

T

1a

1

1

T

T

10

For ak where k ≠ 0, we get

,T

Te

Tjk

1dte

T

1a

1

1tjkT

T 0

tjk

0

1

1

0

k = 0

which we can rewrite as

(3.37a)

(3.37b)


,j2

ee

Tk

2a

1010 TjkTjk

0k

,Tk

)Tksin(2

0

10

k ≠ 0

where 0T = 2. To illustrate the change in the amplitude spectrum |ak|, it turns out to be more appropriate to plot the amplitude spectrum |ak| scaled by T; that is, a plot of T|ak| versus = k0 will be generated.

(3.38)

,k

)Tsin2Ta

0

1k

That is, with thought of as a continuous variable, the function “2sin2sinTT11//” represents the envelopeenvelope of Tak, and the

(3.39)


coefficients ak are simply equally space samples of this envelope. As T1 is fixed, the envelope of Tak is independent of T. The plots of Tak for T=4T1, T=8T1, and T=16T1 are illustrated in Fig. 3.5.

Figure 3.5 The Fourier series coefficients and their envelopeof the periodic square wave in Fig. 3.4 for severalvalues of T (with T1 fixed): (a) T=4T1; (b) T=8T1; (c) T=16T1.


From Fig. 3.5 it can be seen that as T is increased (with T1 fixed), the “density” of the frequency components, ak, increase, whereas the envelope of the magnitudes of the scaled spectral components, Tak, remains the same. In other words, the envelope is sampled with a closer and closer spacing. In the limit as T → ∞, Tak converge into a continuum of frequency components whose magnitudes have the same shape as the envelope of the discrete spectra shown in Fig. 3.5 and the pulse train xT(t) approaches a rectangular pulse, x(t), as shown in Fig. 3.6. That is, in mathematical terms,

,)t(xlim)t(x TT

(3.40)


T1-T1

......

Figure 3.6 A pulse signal as limit of a pulse train signal.

)t(xlim)t(x TT

This example illustrates the effect of extending the period of a periodic signal to infinity which is the basic idea behind the development of Fourier representation for aperiodic signals.

Previously, we have seen the effect of increasing the period of a periodic signal to “density” of the frequency components. Let us now examine the effect of this on the Fourier series representation of a periodic signal.


Let consider an aperiodic signal x(t) that is of finite duration (band limited). That is, for some number T1, x(t) = 0 if |t| > T1, as illustrated in Fig. 3.6 (a pulse signal). Then, we can reconstruct a periodic signal xT(t) for which x(t) is one period, as shown in Fig. 3.4(a pulse train signal). The periodic signal xT

(t) has an exponential Fourier series that is given by

0jk tT k

k

x (t) a e ,

where 0T = 2 and

0

T

2jk t

k TT

2

1a x (t)e dt .

T

(3.41)

(3.42)


As far as the integration is concerned in Eq. 3.42, the integrand on the integral can be rewritten as:

2

T

2

T

tjkk dte)t(x

T

1a 0

Now, let us generalize the formula for the Fourier series coefficient in Eq. 3.43 by defining the envelope X(j) of Tak as

.dte)t(xT

1 tjk 0

(3.43)

dte)t(x)j(X tj

(3.44)


From Eq. 3.48, the signal xT(t) becomes x(t) when T → ∞. That is,

Using Eq. 3.44, we can rewrite Eq. 3.43 as

)jk(XT

1)j(X

T

1a 0k (3.45)

Combining Eq. 3.41 and Eq. 3.45, we can express xT(t) in terms of X(j) as

k

tjk0T ,e)jk(X

T

1)t(x 0

(3.46)

Let = 2 = 2/T/T..

k

tjk0T .e)jk(X

2

1)t(x 0 (3.48)

Then, we can rewrite Eq. 3.46 as:

(3.47)


k

tjk0

T

0e)jk(X2

1lim)t(x (3.49)

From Eq. 3.47, the space between each frequency component, , becomes dense as T → ∞. That is, k0 → , → d, and the summation becomes an integral. Hence in the limit Eq. 3.49 becomes

(3.50)

.de)j(X2

1)t(x tj

X(j) in Eq. 3.44 is referred to as the FouriFourier transformer transform or Fourier integralFourier integral of x(t) and Eq. 3.50 is referred to as the inverse Fourier inverse Fourier transformtransform equation.


3.5.1 The Definition

Let x(t) be a signal such that:

(a) x(t), -∞ < t < ∞, and

(b) for 0 < M < ∞,Mdt|)t(x|

Then the Fourier transformFourier transform of x(t) is defined as

.dte)t(x)j(X)}t(x{ tj

Absolutely integrableAbsolutely integrable

Analysis equation Analysis equation direct transformdirect transform

(3.51)


AndX(j) ↔ x(t)

is called a Fourier transform pairFourier transform pair.

X(jX(j))

.de)j(X2

1)t(x)}j(X{ tj1

(3.52)

SpectrumSpectrum (of signal) (of signal)(a graphical representation of the (a graphical representation of the frequency content of signal)frequency content of signal)

The inverse Fourier transforminverse Fourier transform is defined as

Synthesis equation Synthesis equation inverse transforminverse transform

X(j) provides the information needed for describing x(t) as a linear combination of sinusoidal signals at different frequencies.


We can notice that both Fourier series coefficients, {ak} and Eq. 3.52 represent a signal as a linear combination of complex exponential signals. For periodicperiodic signals, these complex exponential signals have amplitudes {a

k} and occur at a discretediscrete set of harmonically related frequencies k0, k = . For aaperiodiperiodicc signals, these complex exponential signals occur at a continuum continuum of frequencies. That is, the essential difference between the Fourier series and the Fourier transform is that the spectrum in the latter case is continuous. Therefore, the synthesis of an aperiodic signal from its spectrum (Eq. 3.52) is accomplished by means of integration instead of summation.

,...3,2,1,0


3.5.2 Convergence of FourierTransforms

Eq. 3.44 indicates that the definition of the Fourier transform relies on the existence of infinite integrals. Hence we should verify that the infinite integrals in the definition exist for a class of signals before we use the Fourier transform. In this section we simply state several convergence conditions on the signal x(t) since an analysis of convergence is beyond the scope of this class.

The set of conditions that guarantee the existence of the Fourier transform is the DirichlDirichlet conditionset conditions, which may be express as:

1. The signal x(t) is absolutely integrable, that is,


2. The signal x(t) has a finite number of finite discontinuities.

3. The signal x(t) has a finite number of maxima and minima.

The first condition implies that x(t) also has finite energy: that is,

,dt|)t(x| 2

(3.54)

However, the converse is not true. That is, a signal may have finite energy but may not be absolutely integrable. Hence, the finite energy condition of a signal is a weaker condition. In any case, nearly all finite energy signals have a Fourier transform, although, they violate the first condition.

,dt|)t(x|

(3.53)


Thus, the Dirichlet conditions are sufficientbut not necessary for the existence of the Fourier transform.

Fourier transform can be used to represent periodic signals. Previously, we only concentrate on the Fourier transform of an aperiodic signal. In this section, we are going to consider both periodic and aperiodic signals within a unified context.

Let consider a signal x(t) with Fourier transform X(j) that defined as:

X(j) = ( - k0) (3.55)

3.5.3 The Fourier Transform forPeriodic Signals


The signal x(t) having the spectrum defined by Eq. 3.55 can be obtained by taking the inverse Fourier transform; that is,

Hence, the inverse Fourier transform of a single impulse defined by Eq. 3.55 is a complecomplex sinusoidalx sinusoidal. In fact, it is a periodic signal x(t).

Now let consider the case when X(j) represented as a linear combinationa linear combination of

j t0

1x(t) ( k )e d

2

0jk t1e .

2

(3.56)

That is, the spectrum of signal x(t) is a singlsingle impulse at e impulse at = k = k00.


Eq. 3.58 corresponds exactly to the Fourier series representation of a periodic signal. Therefore, the Fourier transform of a periodithe Fourier transform of a periodic signal can be constructed directly from its c signal can be constructed directly from its Fourier seriesFourier series.

k 0k

X(j ) 2 a ( k )

(3.57)

Similarly, the inverse Fourier transform yields

k

tjkk

0ea)t(x (3.58)

scaled impulses equally spaced in frequency, that is,

That is, the spectrum of signal x(t) is now a impulse train of area 2impulse train of area 2aakk at at = k = k00.


3.5.4 Properties of Continuous-TimeFourier Transform

Let x(t) ↔ X(j) and y(t) ↔ Y(j).

Linearity

where a and b are constants.

(3.59)

Time Shifting

0j t0x(t t ) e X( j ) (3.40)

That is, a signal which is shifted in time has only the phasephase of its Fourier transform chachangednged by the factor of “--tt00”.

ax(t) + by(t) ↔ aX(j) + bY(j).


Frequency Shifting

0j t0x(t)e X( j( )) (3.41)

If x(t) is realreal-valued time signal,

*X( j ) X ( j ) (3.43)

That is,

|X(j)| = |X(-j)|

and arg{X(j)} = -arg{X(-j)} (3.44)

or,

Conjugation and Conjugate Symmetry

* *x (t) X ( j ) (3.42)


Re{X(j)} = Re{X(-j)}

and Im{X(j)} = -Im{X(-j)} (3/45)

*X(j ) X ( j ) (3.47)

Eq. 3.47 implies that X(j) is realreal.

Similarly, if x(t) is realreal valued and has oddodd symmetry,

*X(j ) X ( j ) (3.48)

Eq. 3.48 implies that X(j) is imaginaryimaginary.

If x(t) is imaginaryimaginary-valued time signal,

*X( j ) X ( j ) (3.46)

If x(t) is realreal valued and has eveneven symmetry,


Differentiation

dx(t)j X( j )

dt (3.49)

Integration

t 1x( )d X( j ) X(0) ( )

j

(3.50)

Time and Frequency Scaling

1 jx(at) X

| a | a

(3.51)


Duality

(3.52)X(t) 2 x( j( ))

Parseval’s Relation

2 21| x(t) | dt | X( j ) | d

2

(3.53)

Convolution

y(t) x(t) Y( j )X( j ) (3.54)

Time DomainTime Domain


1y(t)x(t) Y( j ) X( j )

2 (4.35)

Frequency DomainFrequency Domain

where

Y(j ) X( j ) Y( j )X( j( ))d

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BYST SigSys - WS2003: Fourier Rep. 120 CPE200 Signals and Systems Chapter 3: Fourier Representations for Signals (Part I)