1 SIGNAL SPECTRA AND EMC One of the critical aspects of sound EMC analysis and design is a basic understanding of signal spectra. A knowledge of the approximate spectral content of common signal types provides the EMC engineer a powerful tool in the identification of possible interference sources. Rather than deal with the exact spectral functions for these basic signal types (which can be somewhat complicated), simple bounds on the signal spectral content will be developed. These bounding functions for signal spectra allow for the EMC analysis in a complicated system to be simplified. Signals can be classified in a variety of ways. Given below are some basic classifications regarding signals and their temporal (time-domain) characteristics or spectral (frequency-domain) characteristics. For all signals, these temporal and spectral characteristics are directly related. Periodic signal - a signal that occurs repetitively over all time t (, ). A periodic signal satisfies the following relationship, where T o defines the period of the signal. The period is related to the signal frequency according to Aperiodic signal - any signal that is not periodic (not repetitive or repetitive only over a subset of time). Deterministic signal - a signal for which the time behavior is precisely defined. Non-deterministic (random) signal - a signal for which the time behavior can only be defined in a statistical sense.
2 Narrowband signal - a signal whose energy is concentrated around a single frequency (sinusoids). Broadband signal - a signal whose energy is not concentrated around 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 power characteristics. Using rms values of voltage v(t) or current i(t), the instantaneous power delivered to a resistive element R can be defined as If we assume a resistance value of unity, the instantaneous power can be defined in terms of a general signal x(t) as The total energy E associated with the signal x(t) over the time interval defined by t (t 1, t 2) is found by integrating the instantaneous power over the given interval.
3 The signal x(t) is defined as an energy signal if the total energy of the signal over the interval of all time is finite. The average power P associated with the signal x(t) is found by dividing the total energy in the interval t (t, t ) by the duration of the interval. 1 2 If the interval is expanded to include all time, the signal x(t) is defined as a power signal if the average power over all time is finite. According to the definition of power and energy signals, a periodic signal has infinite energy but finite average power. Thus, a periodic signal is defined as a power signal. Aperiodic signals that are bounded in time (zero-valued outside some time interval) have finite energy and zero average power. Thus, such an aperiodic signal is defined as an energy signal.
4 PERIODIC SIGNALS AS SERIES EXPANSIONS OF ORTHOGONAL BASIS FUNCTIONS Periodic signals of arbitrary shape can be represented as a series summation of basis functions. The series representation of the arbitrary signal x(t) is defined by where ö n(t) denotes the basis function and c n denotes the corresponding expansion coefficient. The basis functions must be periodic with the same period 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 series representation, it is desirable to have a finite number of terms in the expansion, but this is not always possible. For an expansion requiring an infinite number of terms, a highly convergent series is desirable. With expansion coefficients that rapidly approach zero as the coefficient index increases, the signal x(t) can be accurately approximated by a finite summation with a small number of terms. Also, we may take advantage of the principle of superposition (given certain restrictions) to simplify the system analysis for a particular basis function. The set of basis functions chosen for the series expansion may have a special property known as orthogonality. This property greatly simplifies the determination of the all-important expansion coefficients. Given an orthogonal set of basis functions, then c where * denotes the complex conjugate of the basis function and t is an arbitrary location in time.
5 The orthogonality property defined above can be used to determine the expansion coefficients by multiplying both sides of the series expansion * by ö m (t) and integrating both sides of the equation over one period which gives Solving this equation for the expansion coefficient (and changing the index to n) gives Note that the expansion coefficients for the series expansion are determined by integrating the product of the signal x(t) and the conjugate of the basis function over one period. The use of orthogonal basis functions in the series expansion of the signal has an additional advantage with regard to the number of terms needed in the summation to accurately approximate the signal. Orthogonal basis functions minimize the integral-square approximation error (ISE) when a finite number of terms is used to approximate the signal. The integral square approximation error is given by where the second term in the integrand represents the N-term approximation to the signal x(t). Thus, orthogonal basis functions will yield a lower ISE than non-orthogonal basis functions for any value N.
6 FOURIER SERIES The selection of sinusoidal basis functions in the orthogonal expansion of periodic signals offers two significant advantages in the analysis of EMC problems. (1) Given a linear system, the representation of a periodic signal as a summation of sinusoids allows us to solve the problem via superposition where standard phasor analysis techniques may be applied. (2) The representation of a periodic signal as a summation of sinusoids provides the EMC engineer with a complete picture of the signal spectral content. Combining this information with the knowledge of how antennas radiate sinusoidal signals allows the engineer to more easily identify EMC issues (either in the design process, or in the testing phase). Given a single-input, single-output system as shown below, the system is linear if (1) input x 1(t) produces output y 1(t), input x 2(t) produces output y 2(t), input x 1(t) + x 2(t) produces output y 1(t) + y 2(t). (2) input x(t) produces output y(t), input kx(t) produces output ky(t), where k is a constant. x(t) Linear System y(t)
7 The sinusoidal basis functions associated with the trigonometric Fourier series are where ù o 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 orthogonality relationships among the basis functions.
8 Note that the Fourier series representation of the periodic signal includes terms at integer multiples of the fundamental frequency given by nf o (n = 0, 1, 2... ) The n = 0 term of the Fourier series (associated with the a 0 expansion coefficient) represents the average value of the signal or the signal DC offset. The n = 1 terms of the Fourier series are those associated with the fundamental frequency f o. The n = 2 and higher terms are associated with multiples of the fundamental frequency known as harmonics. Significant spectral energy can be contained in the signal harmonics depending on the time characteristics of the signal.
9 Example Trigonometric Fourier series of a rectangular pulse train (idealized clock signal) The Fourier expansion coefficients of the rectangular pulse train are found by evaluating the appropriate integrals.
10 Inserting the expansion coefficients into the Fourier series for the rectangular pulse train yields The Fourier coefficients of the rectangular pulse train may be normalized by the pulse amplitude and written in terms of the pulse train duty cycle (ô/t ) to yield o We may plot the normalized coefficients vs. the duty cycle of the pulse train to investigate how the energy is distributed in the frequency domain.
12 Note the following special cases and the corresponding Fourier coefficient characteristics. ô/t o 0 (impulse) (a 0, a n, b n) 0 at the same rate, Energy is spread out evenly over all frequencies. ô/t o 1 (DC) a 0 1, (a n, b n) 0, Energy is concentrated in the DC term while the energy in the fundamental frequency and harmonics diminish. ô/t o = 0.5 (50% duty cycle) a 0 0.5, a n 0, b n 0 (n - even), Energy is contained in the fundamental and odd harmonics.
13 The trigonometric Fourier series expansion for a 50% duty cycle rectangular pulse train is The pulse train with a 50% duty cycle has the special property of being an odd function when the DC offset is subtracted from the signal. Thus, this pulse train can be described in terms of a DC component plus only odd harmonics. Duty cycles other than 50% do not satisfy this criterion and require even and odd harmonics in the expansion. The trigonometric Fourier series expansion of a given signal can be expressed in a more compact form using the complex exponential Fourier series. The trigonometric Fourier series can be transformed into the complex exponential form using Euler s identity and related identities Inserting these identities into the trigonometric form of the Fourier series gives Grouping common terms yields By shifting the index on the second summation to negative values of n, we find
14 Based on the definition of the trigonometric Fourier coefficients a n and b n, 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 of the Fourier series is more concise than the trigonometric form of the Fourier series and contains only one coefficient (c n) as opposed to three (a 0, a n, b n). Note that the basis functions and the expansion coefficients in the complex exponential form of the Fourier series are complex-valued, in general. The basis functions and expansion coefficients of the trigonometric form of the Fourier series are real-valued. The coefficients of the trigonometric and complex exponential Fourier series are related by The complex exponential Fourier series is defined in terms of positive and negative frequencies (DC plus positive and negative values of the fundamental frequency and harmonics).
15 The expansion coefficients of the complex exponential Fourier series can be expressed in terms of the coefficients of the trigonometric Fourier series or can be determined directly using the orthogonality property of the complex exponential basis functions. The basis functions for the complex exponential 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 the th th product of the n basis function and the conjugate of the m basis function over one period which yields The expansion coefficients are found by multiplying the signal expansion th by the conjugate of the m basis function and integrating over one period. Solving for the expansion coefficient (and changing the index to n) gives The previously determined relationships between the coefficients of the trigonometric Fourier series and the complex exponential Fourier series can easily be determined using this expression for c n by inserting the Euler s identity expression for the complex exponential in the integral above. Note * that coefficients of index n and n are complex conjugates (c n = c n and * c = c ). n n
16 Example Complex exponential Fourier series of a rectangular pulse train Given the same rectangular pulse train considered for the trigonometric Fourier series expansion (amplitude = A, duty cycle = ô/t o), the complex exponential Fourier expansion coefficients are found by evaluating the following integral. These coefficients can be expressed in terms of a commonly used function known as the sinc function. where the sinc function is defined by Inserting the expression for the expansion coefficients into the series expansion gives
17 SIGNAL SPECTRA Since the expansion coefficients for the complex exponential Fourier series are complex-valued, we must plot the magnitude and phase of the coefficients to characterize these coefficients completely. The complex exponential Fourier series defines a two-sided spectrum with coefficients at positive and negative values of frequency. Given that the coefficients of index n and n for any signal x(t) are complex conjugates, the two-sided magnitude spectrum is an even function while the two-sided phase spectrum is an odd function. Note that the periodic signal spectra (magnitude and phase) are discrete spectra (line spectra) defined at DC and positive and negative multiples of the fundamental frequency. The two-sided spectrum associated with the complex exponential Fourier series can be transformed into a one-sided spectrum (DC plus positive frequencies) in the following way. We first separate the DC term, the positive frequency terms and the negative frequency terms according to Changing the index of the second summation to positive values of n gives According to this equation, the magnitudes of the equivalent one-sided spectrum coefficients (for positive frequencies) are twice that of two-sided spectrum coefficients. The DC term of the one-sided spectrum is equal to that of the two-sided spectrum. Also note that the one-sided phase spectrum is equal to that of the two-sided phase spectrum.
18 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 o o where the phase angle of the sinc function is 0 or 180. According to the previous equations, the individual points along the discrete magnitude spectrum lie on the envelope (replace nf o with f ) defined by Note that sinc(x) is zero-valued where sin(ðx) is zero and x 0. The value of sinc(x) at x = 0 is found by applying L Hospital s rule to the indeterminate form of 0/0. The envelope of the rectangular pulse train two-sided magnitude spectrum is zero valued when
19 Two-sided spectrum (ô/t o = 0.5, A = 1)
20 Two-sided spectrum (ô/t o = 0.1, A = 1)
21 One-sided spectrum (ô/t o = 0.5, A = 1)
22 One-sided spectrum (ô/t o = 0.1, A = 1)
23 EFFICIENT TECHNIQUES FOR THE DETERMINATION OF FOURIER SERIES COEFFICIENTS The direct determination of Fourier series coefficients requires evaluation of integrals involving the signal being expanded [x(t)] and the Fourier basis functions [ö n(t)]. The complexity of the integral evaluation increases with the complexity of the signal. In many cases, the signal x(t) may be accurately represented by a piecewise linear approximation. Given a piecewise linear waveform x(t), certain Fourier series properties may be utilized that eliminate the need to evaluate integrals to determine the expansion coefficients. Linearity If a signal x(t) can be written as a linear combination of two or more functions, the Fourier series of the signal is simply the linear combination of Fourier series for the component functions. Thus, the expansion coefficients for the signal x(t) are simply the linear combination of the expansion coefficients for the component functions.
24 Time Shifting If a signal x(t) is delayed in time by an amount á, the delayed signal is defined by x(t á). According to the definition of the complex exponential Fourier series, the representation of the delayed signal is given by The Fourier coefficients of the delayed signal x(t á) are the coefficients 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 opposite sign. Periodic Train of Unit Impulses The unit impulse function ä(t) (sometimes called the delta function) plays an important role in simplifying the evaluation of Fourier coefficients. The unit impulse function is actually not a function 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 a single point (t = 0) such that ä(t) = 0 for t 0.
25 The expansion coefficients for a unit impulse function can be found using the so-called sifting property associated with the integral of the product of any function and the unit impulse. The range of integration does not have to be over all time to yield the previous result. If the location of the delta function lies anywhere within the range of integration, the result is the same. Consider a periodic train of impulses shown below.
26 The expansion coefficients of the periodic train of unit impulses are given by Note that the same result could be obtained by time shifting an impulse train with impulses located at t = (0, ±T o, ±2T o...). This pulse train yields the following coefficients When these coefficients are time-shifted by the time delay á, the same expansion coefficients are found. Differentiation If the complex exponential Fourier series expansion of an arbitrary signal x(t) is differentiated with respect to time, we find For every additional derivative, the expansion coefficients of the original signal are multiplied by another (jnùot) term. In general, the th k derivative of the signal is given by th Thus, the Fourier coefficients of the k derivative of x(t) are
27 FOURIER EXPANSIONS OF PIECEWISE LINEAR PERIODIC SIGNALS The previously defined properties of Fourier series (linearity, timeshifting, impulse trains, and differentiation) can be combined into a general technique of determining the Fourier expansion coefficients for piecewise linear signals. This technique will provide an efficient technique for estimating the spectral content of digital signals that can be represented by a piecewise linear approximation. The technique for determining the Fourier coefficients of a piecewise linear periodic signal x(t) is defined according to the following steps. (1) Differentiate x(t) with respect to t and separate the resulting first derivative into a sum of two functions: a function containing only impulse functions [x ä1(t)] and a remainder function [x 1(t)] that contains no impulse functions. Note that, at this point, we may easily determine the Fourier coefficients of the portion of the signal x(t) that resulted in impulse functions. (2) Differentiate the remainder function from step (1) with respect to t and separate the result as before. Repeat this step until a zero-valued remainder function is achieved (after N derivatives).
28 (3) Determine the Fourier coefficients associated with the impulse trains generated at each step of the process. The Fourier coefficients of the signal x(t) are determined by applying the differentiation rule to each of the impulse trains yielding Example (Piecewise linear signals/rectangular pulse train) Use the previously defined method to determine the complex exponential Fourier expansion of a rectangular pulse train. The first derivative of the rectangular pulse train results in an impulse train plus a zero-valued remainder function [x (t) = 0]. 1
29 For one period, the impulse train is characterized by an impulse of weight A at t = 0 and an impulse of weight A at t = ô. Thus, the Fourier coefficients of this impulse train are given by The Fourier coefficients of the rectangular pulse train are obtained by dividing the Fourier coefficients of the impulse train (obtained after one derivative) by jnù. o This is the same result found when the coefficients were determined by direct integration. The rectangular pulse train Fourier coefficients were expressed in terms of the sinc function according to
30 Example (Piecewise linear signals) Determine the complex exponential Fourier expansion coefficients of the following piecewise linear signal. The first derivative of x(t) yields the following impulse train x ä1(t) and remainder function x (t). 1
31 The derivative of the remainder function x 1(t) yields the following impulse train x (t) with a remainder function x (t) = 0. ä2 2 The impulse train x ä1(t) consists of an impulse of weight A at t = 0. The impulse train x ä2(t) consists of impulses of weight +A/ô at t = 0 and A/ô at t = ô. The Fourier coefficients of these pulse trains are The Fourier coefficients of the signal x(t) are then
32 APPROXIMATE SPECTRA OF DIGITAL CIRCUIT CLOCK WAVEFORMS Clock signals in digital circuits can be accurately approximated by a piecewise linear waveform as shown below. The clock signal of amplitude A includes a non-zero rise time (ô r) and a non-zero fall time (ô f). The duration of the pulse (ô) is defined as the time between the points on the waveform where the signal value is one-half of the signal amplitude. The Fourier expansion coefficients for this waveform are easily determined using the technique described in the previous section. The first derivative of the x(t) yields no impulse functions and a remainder function x (t) consisting of positive and negative pulses. 1 The derivative of the remainder function x 1(t) yields an impulse train x ä2(t) and a zero-valued remainder function [x (t) = 0]. 2
33 Since the first derivative of the signal yielded no impulse functions, the Fourier coefficients of x(t) can be determined using the Fourier coefficients of the pulse train x ä2(t) alone. The impulse function x ä2(t) consists of the following four impulses: Weight Location Coefficients ä2 The Fourier coefficients of the impulse train x (t) are simply the sum of the Fourier coefficients of the four impulses that make up the pulse train.
35 The Fourier coefficients of the signal x(t) are given by which yields For the special case of ô r = ô f, the expression for the Fourier coefficients simplifies to Thus, the Fourier coefficients of the clock signal with equal rise and fall times varies as the product of two sinc functions. One sinc function depends on the rise/fall time as a percentage of the period while the other sinc function depends on the pulse duration as a percentage of the period. To understand how this affects the spectral content of the clock signal, we focus on the magnitude spectrum. For equal rise/fall times, the magnitude of the clock signal Fourier coefficient c is given by n
36 Note that phase of the clock signal spectral coefficient (ô = ô ) is given by r f The magnitude of the clock signal Fourier coefficient can be written in terms of frequency as Recalling that the magnitude of the coefficients for the one-sided spectrum are two times those of the two-sided spectrum, the magnitude of the onesided spectrum coefficients (n 0) are The envelope of the discrete one-sided magnitude spectrum (replacing nf o by f ) is We normally consider the spectral characteristics of signals using units of db on logarithmic frequency scales (Bode plots). Taking 20log 10 of both sides of the envelope of the discrete one-sided magnitude spectrum (V) gives The envelope of the discrete one-sided magnitude spectrum for the clock signal with ô r = ô f can be determined quickly given an understanding of sinc function behavior on a logarithmic plot. The sinc function can be characterized by simple asymptotes for small and large argument values.
37 On a plot of sinc(x) (db) vs. x on a logarithmic scale, the small argument asymptote is the constant 0 db line while the large argument asymptote is a straight line with a slope of 20 db/decade. The two asymptotes intersect at x = 1/ð which defines the break frequency for the sinc function spectral plot. The three terms defined in the envelope of the of the clock signal one-sided spectrum (ô = ô ) provide the following spectral contributions: r f 20log 10(2Aôf o) 20log10sinc(ôf ) constant level, low frequency limit break frequency at f d = 1/(ðô) 0 db below f, 20 db/decade above f d d 20log10sinc(ôrf ) break frequency at f r = 1/(ðô r) 0 db below f, 20 db/decade above f r r
38 Given ô > ô r, the clock signal one-side spectrum envelope breaks downward from the low frequency limit [20log 10(2Aôf o )] at 20 db/decade at f = fd (the break frequency associated with the pulse duration) and downward at 40 db/decade at f = f r (the break frequency associated with the pulse rise/fall time). It should be noted that the DC level associated with the onesided spectral plot is one-half of the low-frequency limit [20log (Aôf )]. 10 o According to the clock signal spectral envelope shown above, the break frequency associated with the signal rise/fall time (f r) increases as the rise/fall time is decreased. Also, the break frequency associated with the pulse duration (f d) increases as the pulse duration (duty cycle) is decreased. The actual shape of the signal spectrum is dependent on the sinc functions associated with the clock duty cycle and the rise/fall time.
39 Example (Clock signal spectrum - duty cycle variation) Plot the one-sided spectrum of a 1-Volt 1 MHz clock signal with 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.)
40 The clock signal spectrum (in dbìv) is given by The low-frequency asymptotes for the three clock signals are
41 Note that the spectral plot covers a frequency range of f o = 1 MHz (fundamental frequency) up to 100f o = 100 MHz (100 th harmonic). The break frequency associated with pulse duration (f d = 637 khz) lies below the fundamental frequency. Also note that spectral data points on the previous plot are located only at odd harmonics. The vertical scale of the plot must be expanded in 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 with a 50% duty cycle (which has even harmonic coefficients that are identically zero). The energy in the even harmonics of the clock signal increases dramatically as the duty cycle is shifted from 50%.
43 Example (Clock signal spectrum - rise/fall time variation) Plot the one-sided spectrum (dbìv) of a 1-Volt 10 MHz clock signal with 50% duty cycle and a rise/fall time of (a.) 20 ns (b.) 10 ns and (c.) 5 ns. (a.) (b.) (c.)
45 Note that the break frequency associated with the rise/fall time (f r ) increases as the rise/fall time of the clock signal decreases. This means that a clock signal with a sharper rise/fall time contains more spectral energy at higher frequencies. Thus, one simple EMC technique for reducing high frequency radiation due to clock signals in a device under test is to increase the rise/fall time of the clock signal. According to the definition of the trapezoidal clock pulse with ô r = ô f, the area under the clock pulse is Aô, irregardless of the rise/fall time. Therefore, the total energy in these signals is equal. The different areas under the spectral envelopes mean that the energy is distributed differently. The variation in the clock signal spectral envelope with the rise/fall time is a simple shift in the second breakpoint at f along the 20 db/decade line. r
46 Given the critical points on the clock signal spectral envelope (lowd and f r ), we may easily estimate the amplitude of frequency amplitude, f the signal spectrum at a given frequency by utilizing the known slopes of the envelope segments.
47 The slope of the segment on a plot of the spectral coefficients in db verses a log scale in frequency is defined in units of db/decade. Thus, the slope M is defined by Given the value of the spectral coefficient at f 1, the value of the coefficient at f may be written as 2 This concept can be easily applied to the straight segments that form the clock signal spectral envelope. Using units of dbìv, we have
48 Example (Spectral envelope at arbitrary frequency) Determine the value of the spectral envelope (dbìv) at the 11 th harmonic of a 1-Volt 10 MHz clock signal with 50% duty cycle and a rise/fall time of (a) 20 ns (b.) 10 ns and (c.) 5 ns. (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 d r
49 APERIODIC SIGNALS / FOURIER TRANSFORMS The relationship between the spectral characteristics of a periodic signal and an aperiodic signal can be determined by investigating the periodic signal spectral characteristics as the period is allowed to approach infinity. For example, if we consider the piecewise linear clock signal of equal rise and fall times, the envelope of the signal was shown to be dependent on the pulse duration and the pulse rise/fall time. By allowing the period of this signal to approach infinity (a single clock pulse), the spacing 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 series coefficients approach zero as the period approaches infinity. The complex exponential Fourier series for the periodic signal x(t) is defined by where the interval of integration on the expansion coefficients is defined as ( T o/2, T o/2) for convenience in extending the period to include all time. Taking the limit as T o, the coefficients c n approach zero. If we take the limit of cnt o as T o, we find the definition of the Fourier transform [X(ù)] for an aperiodic signal [x(t)]. Discrete spectrum (nù o) Continuous spectrum (ù)
50 The discrete set of expansion coefficients in the Fourier series of a periodic signal coalesce into a continuous function (the Fourier transform) for the aperiodic signal. In order to write the equivalent expansion of the aperiodic signal x(t), we must determine the inverse Fourier transformation. The original form of the Fourier series can be stated in terms of the Fourier transform as follows. Note that the spacing between adjacent frequencies in the Fourier series is the fundamental radian frequency (ù = Äù). o Taking the limit as T o gives Discrete spectrum (n ù o) Continuous spectrum (ù) Discrete spacing (Äù) Continuous distribution (dù) Discrete summation ( ) Continuous summation ( ) c T X(ù) n o
51 The Fourier transformation and the inverse Fourier transformation form the transform pair relating the time-domain and frequency-domain characteristics of an aperiodic signal. The same process used to determine the expansion coefficients of a piecewise linear periodic signal (by differentiating x(t) until only impulse functions remain) can be applied to determining the Fourier transform of an aperiodic signal. The following properties of the Fourier transform are required. Linearity If a signal x(t) can be written as a linear combination of two or more functions, the Fourier transform of the signal is simply the linear combination of Fourier transforms for the component functions.
52 Thus, the Fourier transform for the signal x(t) is simply the linear combination of the Fourier transforms for the component functions. Time Shifting If a signal x(t) is delayed in time by an amount á, the delayed signal is defined by x(t á). According to the definition of the complex exponential Fourier series, the representation of the delayed signal 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 to time, we find
53 For every additional derivative, the Fourier transform the original th signal is multiplied by another (jù) term. In general, the k derivative of the signal is given by th Thus, the Fourier transform pair associated with the k derivative of x(t) is The differentiation technique used to determine the Fourier series coefficients for the piecewise linear clock signal can also be used to determine the Fourier transform of a single piecewise linear clock pulse (as shown below). Assume ô = ô r f The first derivative of the x(t) yields no impulse functions and a remainder function x (t) consisting of a positive and a negative pulse. 1
54 The derivative of the remainder function x 1(t) yields a system of impulses x (t) and a zero-valued remainder function [x (t) = 0]. ä2 2 The function x ä2(t) consists of the following four impulses: Weight Location Transform ä2 The Fourier transform of the function x (t) is simply the sum of the Fourier transforms of the four impulses.
55 Using the following trigonometric identity, X ä2 can be written as The Fourier transform of x(t) is related to X ä2 by which gives The Fourier transform of the single piecewise linear clock pulse (ô f = ô r) is identical in form to the continuous function defining the envelope of the Fourier series coefficients for the periodic piecewise linear clock signal (ô f = ô ). r
56 LINEAR SYSTEM RESPONSE TO PERIODIC AND APERIODIC SIGNALS We have shown that an arbitrary periodic signal can be written as a discrete summation of sinusoids (Fourier series) while an arbitrary aperiodic signal can be written as a continuous sum of sinusoids (Fourier transform). Thus, we may determine the response of a linear system to any arbitrary signal efficiently using phasor analysis (frequency domain). Determination of the system response in the frequency domain is typically more efficient than the solution in the time domain. Consider the linear system 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 of a convolution integral in the time-domain. To determine the system response in the frequency domain, we transform the time domain quantities into the frequency domain (let s = jù).
57 The Fourier transform of the system impulse response h(t) [time domain] is defined as the system transfer function H(s) [frequency domain]. The determination of the system response in the frequency domain involves a simple multiplication of the system input X(s) and the system transfer function H(s). Thus, given an aperiodic input signal, the frequency domain system response is a product of the system transfer function and the Fourier transform of the input signal. If the signal input is periodic and written in terms of a Fourier series, we may apply each individual sinusoid to the system, perform the necessary phasor analysis to determine the resulting output, and sum the outputs to determine the overall system response.
58 Example (Spectra of digital waveforms / linear system response) A 25 MHz clock oscillator (amplitude = 5 V, 50% duty cycle, ô r = ôf = 5 ns) is connected to a logic gate as shown below. A noise suppression capacitor C is placed in parallel with the gate input in th order to reduce the level of the 7 harmonic at the gate input by 20 th db. Determine (a.) the level of the 7 harmonic in the clock output (in dbìv) prior to the placement of the noise suppression capacitor using the exact expression and using spectral bounds (b.) the value of the noise suppression capacitor C. (a.) The exact spectral coefficients of the clock source v S(t) are given by
59 For the seventh harmonic of this clock signal, we have Using spectral bounds, the break frequencies associated with the clock signal pulse duration and rise/fall times are: With f > f r, the approximate spectral coefficients of the clock signal are approximated by The spectral content of the signal at the gate input v G(t) can be related to that of the voltage source v S(t) through simple phasor analysis of the overall circuit.
60 The frequency-domain circuit (phasors, impedances) for the clock/gate circuit is shown below. Using voltage division: The transfer function relating the gate input voltage to the source voltage (prior to the introduction of the noise reduction capacitor) is At the seventh harmonic of the clock signal, we have
61 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 equivalent capacitance is C eq = C G + C. Thus, the transfer function relating the source voltage to the gate input voltage is modified as shown below.
62 In order for the magnitude of the gate input voltage to be reduced by 20 db (one order of magnitude), the transfer function must satisfy Solving for (C G + C) gives
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