SIGNAL SPECTRA AND EMC

One of the critical aspects of sound EMC analysis and design is abasic understanding of signal spectra. A knowledge of the approximatespectral content of common signal types provides the EMC engineer apowerful tool in the identification of possible interference sources. Ratherthan deal with the exact spectral functions for these basic signal types(which can be somewhat complicated), simple bounds on the signal spectralcontent will be developed. These bounding functions for signal spectraallow for the EMC analysis in a complicated system to be simplified.

Signals can be classified in a variety of ways. Given below are somebasic classifications regarding signals and their temporal (time-domain)characteristics or spectral (frequency-domain) characteristics. For allsignals, these temporal and spectral characteristics are directly related.

Periodic signal - a signal that occurs repetitively over all timet 0 (!4,4). A periodic signal satisfies the followingrelationship,

owhere T defines the period of the signal. The period isrelated to the signal frequency according to

Aperiodic signal - any signal that is not periodic (not repetitiveor repetitive only over a subset of time).

Deterministic signal - a signal for which the time behavior isprecisely defined.

Non-deterministic (random) signal - a signal for which the timebehavior can only be defined in a statistical sense.

Narrowband signal - a signal whose energy is concentratedaround a single frequency (sinusoids).

Broadband signal - a signal whose energy is not concentratedaround a single frequency (non-sinusoids).

Examples (signal types)

A pure sinusoid.(periodic, narrowband, deterministic signal)

A clock signal in a digital device.(periodic, broadband, deterministic signal)

A data signal in a digital device.(periodic, broadband, non-deterministic signal)

An electrostatic discharge.(aperiodic, broadband, deterministic signal)

Signals can also be classified with regard to their energy or powercharacteristics. Using rms values of voltage v(t) or current i(t), theinstantaneous power delivered to a resistive element R can be defined as

If we assume a resistance value of unity, the instantaneous power can bedefined in terms of a general signal x(t) as

The total energy E associated with the signal x(t) over the time interval

1 2defined by t 0 (t , t ) is found by integrating the instantaneous power overthe given interval.

The signal x(t) is defined as an energy signal if the total energy of thesignal over the interval of all time is finite.

The average power P associated with the signal x(t) is found by dividing

1 2the total energy in the interval t 0 (t , t ) by the duration of the interval.

If the interval is expanded to include all time, the signal x(t) is defined asa power signal if the average power over all time is finite.

According to the definition of power and energy signals, a periodic signalhas infinite energy but finite average power. Thus, a periodic signal isdefined as a power signal. Aperiodic signals that are bounded in time(zero-valued outside some time interval) have finite energy and zeroaverage power. Thus, such an aperiodic signal is defined as an energysignal.

PERIODIC SIGNALS AS SERIES EXPANSIONS OF

ORTHOGONAL BASIS FUNCTIONS

Periodic signals of arbitrary shape can be represented as a seriessummation of basis functions. The series representation of the arbitrarysignal x(t) is defined by

n nwhere ö (t) denotes the basis function and c denotes the correspondingexpansion coefficient. The basis functions must be periodic with the sameperiod as x(t). A judicious choice of the basis function set can simplify the determination of a system response to the signal x(t). As with any seriesrepresentation, it is desirable to have a finite number of terms in theexpansion, but this is not always possible. For an expansion requiring aninfinite number of terms, a highly convergent series is desirable. Withexpansion coefficients that rapidly approach zero as the coefficient indexincreases, the signal x(t) can be accurately approximated by a finitesummation with a small number of terms. Also, we may take advantage ofthe principle of superposition (given certain restrictions) to simplify thesystem analysis for a particular basis function.

The set of basis functions chosen for the series expansion may havea special property known as orthogonality. This property greatly simplifiesthe determination of the all-important expansion coefficients. Given anorthogonal set of basis functions, then

cwhere “*” denotes the complex conjugate of the basis function and t is anarbitrary location in time.

The orthogonality property defined above can be used to determinethe expansion coefficients by multiplying both sides of the series expansion

mby ö (t) and integrating both sides of the equation over one period which*

gives

Solving this equation for the expansion coefficient (and changing the indexto n) gives

Note that the expansion coefficients for the series expansion are determinedby integrating the product of the signal x(t) and the conjugate of the basisfunction over one period.

The use of orthogonal basis functions in the series expansion of thesignal has an additional advantage with regard to the number of termsneeded in the summation to accurately approximate the signal. Orthogonalbasis functions minimize the integral-square approximation error (ISE) when a finite number of terms is used to approximate the signal. Theintegral square approximation error is given by

where the second term in the integrand represents the N-termapproximation to the signal x(t). Thus, orthogonal basis functions willyield a lower ISE than non-orthogonal basis functions for any value N.

FOURIER SERIES

The selection of sinusoidal basis functions in the orthogonalexpansion of periodic signals offers two significant advantages in theanalysis of EMC problems.

(1) Given a linear system, the representation of a periodic signal asa summation of sinusoids allows us to solve the problem viasuperposition where standard phasor analysis techniques maybe applied.

(2) The representation of a periodic signal as a summation ofsinusoids provides the EMC engineer with a complete pictureof the signal spectral content. Combining this information withthe knowledge of how antennas radiate sinusoidal signalsallows the engineer to more easily identify EMC issues (eitherin the design process, or in the testing phase).

Given a single-input, single-output system as shown below, the system islinear if

1 1(1) input x (t) produces output y (t),

2 2input x (t) produces output y (t),

1 2 1 2input x (t) + x (t) produces output y (t) + y (t).(2) input x(t) produces output y(t),

input kx(t) produces output ky(t), where k is a constant.

x(t) y(t)Linear System

The sinusoidal basis functions associated with the trigonometricFourier series are

owhere ù is the fundamental frequency (in radians) of the periodic signal defined by

The trigonometric Fourier series expansion of the general signal x(t) is

where the expansion coefficients are given by

The expansion coefficients are determined by utilizing the orthogonalityrelationships among the basis functions.

Note that the Fourier series representation of the periodic signal includesterms at integer multiples of the fundamental frequency given by

onf (n = 0, 1, 2 ... )

0The n = 0 term of the Fourier series (associated with the a expansioncoefficient) represents the average value of the signal or the signal DCoffset. The n = 1 terms of the Fourier series are those associated with the

ofundamental frequency f . The n = 2 and higher terms are associated withmultiples of the fundamental frequency known as harmonics. Significantspectral energy can be contained in the signal harmonics depending on thetime characteristics of the signal.

Example Trigonometric Fourier series of a rectangular pulse train (idealized clock signal)

The Fourier expansion coefficients of the rectangular pulse train arefound by evaluating the appropriate integrals.

Inserting the expansion coefficients into the Fourier series for therectangular pulse train yields

The Fourier coefficients of the rectangular pulse train may benormalized by the pulse amplitude and written in terms of the pulse

otrain duty cycle (ô/T ) to yield

We may plot the normalized coefficients vs. the duty cycle of thepulse train to investigate how the energy is distributed in thefrequency domain.

Note the following special cases and the corresponding Fourier coefficientcharacteristics.

n no 0ô/T 6 0 (impulse) (aN , aN , bN ) 6 0 at the same rate,Energy is spread out evenly overall frequencies.

n no 0ô/T 6 1 (DC) aN 6 1, (aN , bN ) 6 0,Energy is concentrated in the DCterm while the energy in thefundamental frequency andharmonics diminish.

n no 0ô/T = 0.5 (50% duty cycle) aN 6 0.5, aN 6 0, bN 6 0 (n -even), Energy is contained in thefundamental and odd harmonics.

The trigonometric Fourier series expansion for a 50% duty cyclerectangular pulse train is

The pulse train with a 50% duty cycle has the special property of being anodd function when the DC offset is subtracted from the signal. Thus, thispulse train can be described in terms of a DC component plus only oddharmonics. Duty cycles other than 50% do not satisfy this criterion andrequire even and odd harmonics in the expansion.

The trigonometric Fourier series expansion of a given signal can beexpressed in a more compact form using the complex exponential Fourierseries. The trigonometric Fourier series can be transformed into thecomplex exponential form using Euler’s identity

and related identities

Inserting these identities into the trigonometric form of the Fourier seriesgives

Grouping common terms yields

By shifting the index on the second summation to negative values of n, wefind

n nBased on the definition of the trigonometric Fourier coefficients a and b ,the coefficients of negative index are related to those of positive index by

The expansion for x(t) can then be written as

The expansion for the signal x(t) above can be written as

which is the complex exponential form of the Fourier series. This form ofthe Fourier series is more concise than the trigonometric form of the

n 0Fourier series and contains only one coefficient (c ) as opposed to three (a ,

n na , b ). Note that the basis functions and the expansion coefficients in thecomplex exponential form of the Fourier series are complex-valued, ingeneral. The basis functions and expansion coefficients of thetrigonometric form of the Fourier series are real-valued. The coefficientsof the trigonometric and complex exponential Fourier series are related by

The complex exponential Fourier series is defined in terms of positive andnegative frequencies (DC plus positive and negative values of the fundamental frequency and harmonics).

The expansion coefficients of the complex exponential Fourier seriescan be expressed in terms of the coefficients of the trigonometric Fourierseries or can be determined directly using the orthogonality property of thecomplex exponential basis functions. The basis functions for the complexexponential Fourier series are defined by

so that the expansion of the arbitrary signal x(t) may be written as

The orthogonality of the basis functions can be shown by integrating theproduct of the n basis function and the conjugate of the m basis functionth th

over one period which yields

The expansion coefficients are found by multiplying the signal expansionby the conjugate of the m basis function and integrating over one period.th

Solving for the expansion coefficient (and changing the index to n) gives

The previously determined relationships between the coefficients of thetrigonometric Fourier series and the complex exponential Fourier series can

neasily be determined using this expression for c by inserting the Euler’sidentity expression for the complex exponential in the integral above. Note

n !nthat coefficients of index n and !n are complex conjugates (c = c and *

n !nc = c ).*

Example Complex exponential Fourier series of a rectangular pulse train

Given the same rectangular pulse train considered for thetrigonometric Fourier series expansion (amplitude = A, duty cycle =

oô/T ), the complex exponential Fourier expansion coefficients arefound by evaluating the following integral.

These coefficients can be expressed in terms of a commonly usedfunction known as the sinc function.

where the sinc function is defined by

Inserting the expression for the expansion coefficients into the seriesexpansion gives

SIGNAL SPECTRA

Since the expansion coefficients for the complex exponential Fourierseries are complex-valued, we must plot the magnitude and phase of thecoefficients to characterize these coefficients completely. The complexexponential Fourier series defines a two-sided spectrum with coefficientsat positive and negative values of frequency. Given that the coefficients ofindex n and !n for any signal x(t) are complex conjugates, the two-sidedmagnitude spectrum is an even function while the two-sided phasespectrum is an odd function. Note that the periodic signal spectra(magnitude and phase) are discrete spectra (line spectra) defined at DC andpositive and negative multiples of the fundamental frequency.

The two-sided spectrum associated with the complex exponentialFourier series can be transformed into a one-sided spectrum (DC pluspositive frequencies) in the following way. We first separate the DC term,the positive frequency terms and the negative frequency terms accordingto

Changing the index of the second summation to positive values of n gives

According to this equation, the magnitudes of the equivalent one-sidedspectrum coefficients (for positive frequencies) are twice that of two-sidedspectrum coefficients. The DC term of the one-sided spectrum is equal tothat of the two-sided spectrum. Also note that the one-sided phasespectrum is equal to that of the two-sided phase spectrum.

Example (One-sided and two-sided spectra of a rectangular pulse train)

For the rectangular pulse train coefficients given by

the magnitude and phase of these coefficients are

where the phase angle of the sinc function is 0 or 180 . Accordingo o

to the previous equations, the individual points along the discrete

omagnitude spectrum lie on the envelope (replace nf with f ) definedby

Note that sinc(x) is zero-valued where sin(ðx) is zero and x �0. Thevalue of sinc(x) at x = 0 is found by applying L’Hospital’s rule to theindeterminate form of 0/0.

The envelope of the rectangular pulse train two-sided magnitudespectrum is zero valued when

oTwo-sided spectrum (ô/T = 0.5, A = 1)

oTwo-sided spectrum (ô/T = 0.1, A = 1)

oOne-sided spectrum (ô/T = 0.5, A = 1)

oOne-sided spectrum (ô/T = 0.1, A = 1)

EFFICIENT TECHNIQUES FOR THE DETERMINATION

OF FOURIER SERIES COEFFICIENTS

The direct determination of Fourier series coefficients requiresevaluation of integrals involving the signal being expanded [x(t)] and the

nFourier basis functions [ö (t)]. The complexity of the integral evaluationincreases with the complexity of the signal. In many cases, the signal x(t)may be accurately represented by a piecewise linear approximation. Givena piecewise linear waveform x(t), certain Fourier series properties may beutilized that eliminate the need to evaluate integrals to determine theexpansion coefficients.

Linearity

If a signal x(t) can be written as a linear combination of two ormore functions,

the Fourier series of the signal is simply the linear combination ofFourier series for the component functions.

Thus, the expansion coefficients for the signal x(t) are simply thelinear combination of the expansion coefficients for the componentfunctions.

Time Shifting

If a signal x(t) is delayed in time by an amount á, the delayedsignal is defined by x(t!á). According to the definition of thecomplex exponential Fourier series, the representation of the delayedsignal is given by

The Fourier coefficients of the delayed signal x(t!á) are thecoefficients of the original signal multiplied by the delay factor of

The same relationship holds true for a signal advanced in time x(t+á)where the argument of the complex exponential term is of oppositesign.

Periodic Train of Unit Impulses

The unit impulse function ä(t) (sometimes called the “delta”function) plays an important role in simplifying the evaluation ofFourier coefficients. The unit impulse function is actually not afunction in the strict mathematical definition, but a distribution. Conceptually, the area under the unit impulse curve is unity such that

but the entire weight of the function is assumed to be located at asingle point (t = 0) such that ä(t) = 0 for t �0.

The expansion coefficients for a unit impulse function can be foundusing the so-called sifting property associated with the integral of theproduct of any function and the unit impulse.

The range of integration does not have to be over all time to yield theprevious result. If the location of the delta function lies anywherewithin the range of integration, the result is the same.

Consider a periodic train of impulses shown below.

The expansion coefficients of the periodic train of unit impulses aregiven by

Note that the same result could be obtained by time shifting an

o oimpulse train with impulses located at t = (0, ±T , ±2T ...). Thispulse train yields the following coefficients

When these coefficients are time-shifted by the time delay á, the sameexpansion coefficients are found.

Differentiation

If the complex exponential Fourier series expansion of anarbitrary signal x(t) is differentiated with respect to time, we find

For every additional derivative, the expansion coefficients of the

ooriginal signal are multiplied by another (jnù t) term. In general, thek derivative of the signal is given by th

Thus, the Fourier coefficients of the k derivative of x(t) are th

FOURIER EXPANSIONS OF PIECEWISE

LINEAR PERIODIC SIGNALS

The previously defined properties of Fourier series (linearity, time-shifting, impulse trains, and differentiation) can be combined into a generaltechnique of determining the Fourier expansion coefficients for piecewiselinear signals. This technique will provide an efficient technique for estimating the spectral content of digital signals that can be represented bya piecewise linear approximation.

The technique for determining the Fourier coefficients of a piecewiselinear periodic signal x(t) is defined according to the following steps.

(1) Differentiate x(t) with respect to t and separate the resulting firstderivative into a sum of two functions: a function containing

ä1 1only impulse functions [x (t)] and a remainder function [x (t)]that contains no impulse functions.

Note that, at this point, we may easily determine the Fouriercoefficients of the portion of the signal x(t) that resulted inimpulse functions.

(2) Differentiate the remainder function from step (1) with respectto t and separate the result as before. Repeat this step until azero-valued remainder function is achieved (after Nderivatives).

(3) Determine the Fourier coefficients associated with the impulsetrains generated at each step of the process.

The Fourier coefficients of the signal x(t) are determined byapplying the differentiation rule to each of the impulse trainsyielding

Example (Piecewise linear signals/rectangular pulse train)

Use the previously defined method to determine the complexexponential Fourier expansion of a rectangular pulse train.

The first derivative of the rectangular pulse train results in an impulse

1train plus a zero-valued remainder function [x (t) = 0].

For one period, the impulse train is characterized by an impulse ofweight A at t = 0 and an impulse of weight !A at t = ô. Thus, theFourier coefficients of this impulse train are given by

The Fourier coefficients of the rectangular pulse train are obtained bydividing the Fourier coefficients of the impulse train (obtained after

oone derivative) by jnù .

This is the same result found when the coefficients were determinedby direct integration. The rectangular pulse train Fourier coefficientswere expressed in terms of the sinc function according to

Example (Piecewise linear signals)

Determine the complex exponential Fourier expansion coefficients ofthe following piecewise linear signal.

ä1The first derivative of x(t) yields the following impulse train x (t) and

1remainder function x (t).

1The derivative of the remainder function x (t) yields the following

ä2 2impulse train x (t) with a remainder function x (t) = 0.

ä1The impulse train x (t) consists of an impulse of weight !A at t = 0.

ä2The impulse train x (t) consists of impulses of weight +A/ô at t = 0and !A/ô at t = ô. The Fourier coefficients of these pulse trains are

The Fourier coefficients of the signal x(t) are then

APPROXIMATE SPECTRA OF DIGITAL CIRCUIT

CLOCK WAVEFORMS

Clock signals in digital circuits can be accurately approximated by apiecewise linear waveform as shown below. The clock signal of amplitude

r fA includes a non-zero rise time (ô ) and a non-zero fall time (ô ). Theduration of the pulse (ô) is defined as the time between the points on thewaveform where the signal value is one-half of the signal amplitude.

The Fourier expansion coefficients for this waveform are easilydetermined using the technique described in the previous section. The firstderivative of the x(t) yields no impulse functions and a remainder function

1x (t) consisting of positive and negative pulses.

1 ä2The derivative of the remainder function x (t) yields an impulse train x (t)

2and a zero-valued remainder function [x (t) = 0].

Since the first derivative of the signal yielded no impulse functions, theFourier coefficients of x(t) can be determined using the Fourier coefficients

ä2 ä2of the pulse train x (t) alone. The impulse function x (t) consists of thefollowing four impulses:

Weight Location Coefficients

ä2The Fourier coefficients of the impulse train x (t) are simply the sum of theFourier coefficients of the four impulses that make up the pulse train.

The Fourier coefficients of the signal x(t) are given by

which yields

r fFor the special case of ô = ô , the expression for the Fourier coefficientssimplifies to

Thus, the Fourier coefficients of the clock signal with equal rise and falltimes varies as the product of two sinc functions. One sinc functiondepends on the rise/fall time as a percentage of the period while the othersinc function depends on the pulse duration as a percentage of the period. To understand how this affects the spectral content of the clock signal, wefocus on the magnitude spectrum. For equal rise/fall times, the magnitude

nof the clock signal Fourier coefficient c is given by

r fNote that phase of the clock signal spectral coefficient (ô = ô ) is given by

The magnitude of the clock signal Fourier coefficient can be written interms of frequency as

Recalling that the magnitude of the coefficients for the one-sided spectrumare two times those of the two-sided spectrum, the magnitude of the one-sided spectrum coefficients (n �0) are

oThe envelope of the discrete one-sided magnitude spectrum (replacing nfby f ) is

We normally consider the spectral characteristics of signals using units of

10dB on logarithmic frequency scales (Bode plots). Taking 20log of bothsides of the envelope of the discrete one-sided magnitude spectrum (V)gives

The envelope of the discrete one-sided magnitude spectrum for the clock

r fsignal with ô = ô can be determined quickly given an understanding of sincfunction behavior on a logarithmic plot. The sinc function can becharacterized by simple asymptotes for small and large argument values.

On a plot of *sinc(x)* (dB) vs. x on a logarithmic scale, the small argumentasymptote is the constant 0 dB line while the large argument asymptote isa straight line with a slope of !20 dB/decade. The two asymptotes intersectat x = 1/ð which defines the break frequency for the sinc function spectralplot.

The three terms defined in the envelope of the of the clock signal one-sided

r fspectrum (ô = ô ) provide the following spectral contributions:

10 o20log (2Aôf ) constant level, low frequency limit

d1020log sinc(ôf ) break frequency at f = 1/(ðô)

d d0 dB below f , !20 dB/decade above f

r r r1020log sinc(ô f ) break frequency at f = 1/(ðô )

r r0 dB below f , !20 dB/decade above f

rGiven ô > ô , the clock signal one-side spectrum envelope breaks downward

d10 ofrom the low frequency limit [20log (2Aôf )] at !20 dB/decade at f = f(the break frequency associated with the pulse duration) and downward at

r!40 dB/decade at f = f (the break frequency associated with the pulserise/fall time). It should be noted that the DC level associated with the one-

10 osided spectral plot is one-half of the low-frequency limit [20log (Aôf )].

According to the clock signal spectral envelope shown above, the break

rfrequency associated with the signal rise/fall time ( f ) increases as therise/fall time is decreased. Also, the break frequency associated with the

dpulse duration ( f ) increases as the pulse duration (duty cycle) is decreased.The actual shape of the signal spectrum is dependent on the sinc functionsassociated with the clock duty cycle and the rise/fall time.

Example (Clock signal spectrum - duty cycle variation)

Plot the one-sided spectrum of a 1-Volt 1 MHz clock signalwith a rise/fall time of 20 ns and a duty cycle of (a.) 50% (b.)30% and (c.) 10%. Plot the spectra in units of dBìV.

(a.)

(b.)

(c.)

The clock signal spectrum (in dBìV) is given by

The low-frequency asymptotes for the three clock signals are

oNote that the spectral plot covers a frequency range of f = 1

oMHz (fundamental frequency) up to 100f = 100 MHz (100th

harmonic). The break frequency associated with pulse duration

d(f = 637 kHz) lies below the fundamental frequency. Also notethat spectral data points on the previous plot are located only atodd harmonics. The vertical scale of the plot must be expandedin order to view the even harmonic spectral data points.

There is very little energy in the even harmonics because this clock signal approximates the ideal rectangular pulse train witha 50% duty cycle (which has even harmonic coefficients thatare identically zero). The energy in the even harmonics of theclock signal increases dramatically as the duty cycle is shiftedfrom 50%.

Example (Clock signal spectrum - rise/fall time variation)

Plot the one-sided spectrum (dBìV) of a 1-Volt 10 MHz clocksignal with 50% duty cycle and a rise/fall time of (a.) 20 ns (b.)10 ns and (c.) 5 ns.

(a.)

(b.)

(c.)

Note that the break frequency associated with the rise/fall time

r(f ) increases as the rise/fall time of the clock signal decreases. This means that a clock signal with a sharper rise/fall timecontains more spectral energy at higher frequencies. Thus, onesimple EMC technique for reducing high frequency radiationdue to clock signals in a device under test is to increase therise/fall time of the clock signal. According to the definition of

r fthe trapezoidal clock pulse with ô = ô , the area under the clockpulse is Aô, irregardless of the rise/fall time. Therefore, thetotal energy in these signals is equal. The different areas underthe spectral envelopes mean that the energy is distributeddifferently. The variation in the clock signal spectral envelopewith the rise/fall time is a simple shift in the second breakpoint

rat f along the !20 dB/decade line.

Given the critical points on the clock signal spectral envelope (low-

d rfrequency amplitude, f and f ), we may easily estimate the amplitude ofthe signal spectrum at a given frequency by utilizing the known slopes ofthe envelope segments.

The slope of the segment on a plot of the spectral coefficients in dB versesa log scale in frequency is defined in units of dB/decade. Thus, the slopeM is defined by

1Given the value of the spectral coefficient at f , the value of the coefficient

2at f may be written as

This concept can be easily applied to the straight segments that form theclock signal spectral envelope. Using units of dBìV, we have

Example (Spectral envelope at arbitrary frequency)

Determine the value of the spectral envelope (dBìV) at the 11th

harmonic of a 1-Volt 10 MHz clock signal with 50% duty cycleand a rise/fall time of (a) 20 ns (b.) 10 ns and (c.) 5 ns.

d r(a.) f = 6.37 MHz, f = 15.9 MHz

d r(b.) f = 6.37 MHz, f = 31.8 MHz

d r(c.) f = 6.37 MHz, f = 63.7 MHz

APERIODIC SIGNALS / FOURIER TRANSFORMS

The relationship between the spectral characteristics of a periodicsignal and an aperiodic signal can be determined by investigating theperiodic signal spectral characteristics as the period is allowed to approachinfinity. For example, if we consider the piecewise linear clock signal ofequal rise and fall times, the envelope of the signal was shown to bedependent on the pulse duration and the pulse rise/fall time. By allowingthe period of this signal to approach infinity (a single clock pulse), thespacing between the adjacent terms in the Fourier series approaches zero.Thus, the discrete Fourier series coalesces into a continuous spectrum(defining the Fourier transform). The magnitude of the Fourier seriescoefficients approach zero as the period approaches infinity.

The complex exponential Fourier series for the periodic signal x(t) isdefined by

where the interval of integration on the expansion coefficients is defined as

o o(!T /2, T /2) for convenience in extending the period to include all time.

noTaking the limit as T 6 4, the coefficients c approach zero. If we take the

n o olimit of c T as T 6 4, we find the definition of the Fourier transform[X(ù)] for an aperiodic signal [x(t)].

oDiscrete spectrum (nù ) 6 Continuous spectrum (ù)

The discrete set of expansion coefficients in the Fourier series of aperiodic signal coalesce into a continuous function (the Fourier transform)for the aperiodic signal. In order to write the equivalent “expansion” of theaperiodic signal x(t), we must determine the inverse Fourier transformation. The original form of the Fourier series can be stated in terms of the Fouriertransform as follows.

Note that the spacing between adjacent frequencies in the Fourier series is

othe fundamental radian frequency (ù = Äù).

oTaking the limit as T 6 4 gives

oDiscrete spectrum (n ù ) 6 Continuous spectrum (ù)

Discrete spacing (Äù) 6 Continuous distribution (dù)

Discrete summation (G) 6 Continuous summation (I)

n o c T 6 X(ù)

The Fourier transformation and the inverse Fourier transformation form the“transform pair” relating the time-domain and frequency-domaincharacteristics of an aperiodic signal.

The same process used to determine the expansion coefficients of apiecewise linear periodic signal (by differentiating x(t) until only impulsefunctions remain) can be applied to determining the Fourier transform ofan aperiodic signal. The following properties of the Fourier transform arerequired.

Linearity

If a signal x(t) can be written as a linear combination of two ormore functions,

the Fourier transform of the signal is simply the linear combinationof Fourier transforms for the component functions.

Thus, the Fourier transform for the signal x(t) is simply the linearcombination of the Fourier transforms for the component functions.

Time Shifting

If a signal x(t) is delayed in time by an amount á, the delayedsignal is defined by x(t!á). According to the definition of thecomplex exponential Fourier series, the representation of the delayedsignal is given by

Thus, the Fourier transform pair for a time-shifted signal is given by

Fourier Transform of a Unit Impulse

The Fourier transform of a unit impulse located at t = á [ä(t!á)]is given by

where the sifting property of the impulse function has been utilized.

Differentiation

If the inverse Fourier transform is differentiated with respect totime, we find

For every additional derivative, the Fourier transform the originalsignal is multiplied by another (jù) term. In general, the k derivativeth

of the signal is given by

Thus, the Fourier transform pair associated with the k derivative ofth

x(t) is

The differentiation technique used to determine the Fourier seriescoefficients for the piecewise linear clock signal can also be used todetermine the Fourier transform of a single piecewise linear clock pulse (asshown below).

r f Assume ô = ô

The first derivative of the x(t) yields no impulse functions and a remainder

1function x (t) consisting of a positive and a negative pulse.

1The derivative of the remainder function x (t) yields a system of impulses

ä2 2x (t) and a zero-valued remainder function [x (t) = 0].

ä2The function x (t) consists of the following four impulses:

Weight Location Transform

ä2The Fourier transform of the function x (t) is simply the sum of the Fouriertransforms of the four impulses.

Using the following trigonometric identity,

ä2X can be written as

ä2The Fourier transform of x(t) is related to X by

which gives

f rThe Fourier transform of the single piecewise linear clock pulse (ô = ô ) isidentical in form to the continuous function defining the envelope of the

fFourier series coefficients for the periodic piecewise linear clock signal (ô

r= ô ).

LINEAR SYSTEM RESPONSE TO PERIODIC

AND APERIODIC SIGNALS

We have shown that an arbitrary periodic signal can be written as adiscrete summation of sinusoids (Fourier series) while an arbitraryaperiodic signal can be written as a continuous sum of sinusoids (Fouriertransform). Thus, we may determine the response of a linear system to anyarbitrary signal efficiently using phasor analysis (frequency domain). Determination of the system response in the frequency domain is typicallymore efficient than the solution in the time domain. Consider the linearsystem shown below with input x(t), output y(t), and impulse response h(t). The system response y(t) in terms of the system input x(t) takes the form ofa convolution integral in the time-domain.

To determine the system response in the frequency domain, we transformthe time domain quantities into the frequency domain (let s = jù).

The Fourier transform of the system impulse response h(t) [time domain]is defined as the system transfer function H(s) [frequency domain]. Thedetermination of the system response in the frequency domain involves asimple multiplication of the system input X(s) and the system transferfunction H(s).

Thus, given an aperiodic input signal, the frequency domain systemresponse is a product of the system transfer function and the Fouriertransform of the input signal. If the signal input is periodic and written interms of a Fourier series, we may apply each individual sinusoid to thesystem, perform the necessary phasor analysis to determine the resultingoutput, and sum the outputs to determine the overall system response.

Example (Spectra of digital waveforms / linear system response)

r fA 25 MHz clock oscillator (amplitude = 5 V, 50% duty cycle, ô = ô= 5 ns) is connected to a logic gate as shown below. A noisesuppression capacitor C is placed in parallel with the gate input inorder to reduce the level of the 7 harmonic at the gate input by 20th

dB. Determine (a.) the level of the 7 harmonic in the clock outputth

(in dBìV) prior to the placement of the noise suppression capacitorusing the exact expression and using spectral bounds (b.) the value ofthe noise suppression capacitor C.

S(a.) The exact spectral coefficients of the clock source v (t) aregiven by

For the seventh harmonic of this clock signal, we have

Using spectral bounds, the break frequencies associated withthe clock signal pulse duration and rise/fall times are:

rWith f > f , the approximate spectral coefficients of the clocksignal are approximated by

GThe spectral content of the signal at the gate input v (t) can be

Srelated to that of the voltage source v (t) through simple phasoranalysis of the overall circuit.

The frequency-domain circuit (phasors, impedances) for theclock/gate circuit is shown below.

Using voltage division:

The transfer function relating the gate input voltage to thesource voltage (prior to the introduction of the noise reductioncapacitor) is

At the seventh harmonic of the clock signal, we have

The magnitudes of the source voltage and the gate input voltage(in dBìV) at the seventh harmonic are related by

From previous results,

so that

(b.) The noise suppression capacitor C, when added to the circuit,is connected in parallel with the gate capacitance. The resulting

eq Gequivalent capacitance is C = C + C. Thus, the transferfunction relating the source voltage to the gate input voltage ismodified as shown below.

In order for the magnitude of the gate input voltage to bereduced by 20 dB (one order of magnitude), the transferfunction must satisfy

GSolving for (C + C) gives