Signals can be classified according to attributes. A few such classifications are outlined

Transcription

1 Chapter : Signal and Linear System Analysis Signals can be classified according to attributes. A few such classifications are outlined below. ) A deterministic signal can be specified as a function of time by a mathematical formula. At any instant of time t, x(t) is a number. For example, x(t) = Acost is a deterministic signal.. Random signals take on random values at any given time; at time t, random signal x(t) is a random variable, not a simple number. Random signals must be modeled probabilistically using random process theory. 3. Periodic signals are deterministic signals for which x(t) x(t T), - <t<, (-) where T is a constant. The smallest such T is called the period of x. 4. Aperiodic signals are deterministic signals that are not periodic. Usually, they are nonzero only over a finite length of time. The commonly-used single pulse is a simple example of an aperiodic signal. Dirac Delta Function The Dirac Delta function is denoted as (t), and it is defined by the property x(t) (t)dt x(), (-) x(t) y(t) A A -T T T T t t a) b) Figure -: a) Periodic signal with period T. b) Aperiodic signal. Latest Updates at -

2 where x(t) is any function that is continuous at the origin. The delta function is only defined in terms of what it does to integrals (it only appears under an integral sign). In (-), a simple change of variable produces x(t) (t t )dt x(t ), (-3) where it is assumed that x(t) is continuous at t. Formula (-3) is called the sifting property of the delta function. Other properties of the delta function are (at) (t), a, (-4) a ( t) (t), (-5) x(t ), t t t t x(t) (t t t )dt (-6), otherwise, x(t) (t t ) x(t ) (t t ), and (-7) t (n) n (n) (-8) t x(t) (t t )dt ( ) x (t ), t t t. In (-8), the quantity x (n) (t) denotes the n th derivative of x; function x(t) and its first n derivatives are assumed to be continuous at t = t. (n) is often called the n th generalized derivative of The delta function can be thought of as the limit of a suitably chosen conventional function having unity area in an infinitesimally small width. Figure - illustrates three commonly used approximations. The first is a rectangular pulse of width and height of /. It behaves like a delta function because Latest Updates at -

3 (t) (t) sin( t / ) t / / a) Width b) t 5 x (x) exp =. 4 (x) 3 = x-axis c) Figure -: Three commonly-used approximations to the delta function. limit (t)x(t) dt limit ( ) /( ) x() x() (-9) Latest Updates at -3

4 for any x(t) that is continuous at t =. The approximation depicted by Figure -b is given by (t) sin( t / ) Sa( t / ) t, (-) where Sa(x) = {sin(x)}/x. As, function (-) becomes a delta function. Finally, Figure -c depicts the approximation t (t) exp. (-) All three approximations are of unit area, independent of >. Unit Step Function The unit step function is denoted as U(t), and it is defined as, t U(t). (-) t The unit step is related to the delta function by the relationship t U(t) (x)dx (-3) du(t) (t). (-4) dt The unit step has many uses. For example, it can be used to construct complicated functions from simple pieces. Figure -3 illustrates a function that can be constructed from delayed ramps. Unit step functions can be used to describe this function as Latest Updates at -4

5 x(t) 3 time Figure -3: Simple signal that can be synthesized using ramps. U(t) tu(t) (t )U(t ) (t )U(t ) (t 3)U(t 3). (-5) Energy and Power Signals The total energy on a per-ohm basis in the complex-valued signal x(t) is defined as E limit T T T x(t) dt. (-6) The total power on a per-ohm basis in the complex-valued signal x(t) is defined as P limit x(t) dt T T T. (-7) T x(t) is said to be an energy signal if < E < (P = for an energy signal). And, x(t) is said to be a power signal if < P < (E = for a power signal). Example -: Consider x(t) = Ae -t U(t), >, where A and are constants. t A E x (t)dt A e dt (-8) x(t) is an energy signal. Example -: Consider s(t) = Acos(t + ), where A,, and are constants. Compute Latest Updates at -5

6 T A T T T P limit A cos ( t ) dt. (-9) x(t) is a power signal since < P <. Generalized Fourier Series Vector space concepts can be applied to signals. Recall the n-dimensional vector space R n. A vector x R n can be represented as n x kk k, (-) where vectors k, k n, represent a basis for the space, and the k are constants. This basic idea extends to time-varying signals defined on an interval [, T]. One can show that a vector space, called L [, T] here, can be defined which consists of all complex-valued signals x(t), t T, that satisfy T x(t) dt. (-) We use as scalars the field of complex numbers. Let k (t), k n, be a set of elements of space L [, T]. Furthermore, require that they be orthogonal, which means T k(t) j(t)dt c k, k j, (-) k j where the c k are real (nonzero) numbers. The orthogonal k (t), k n, are said to orthonormal if c k =, k n. Let x(t) be an arbitrary signal (i.e., vector) in L [, T]. Approximate x by the Latest Updates at -6

7 sum n x a(t) k k(t) k, (-3) where k are scalars (complex numbers). Pick the k to minimize the integral square error (ISE) n T T en x a(t) x(t) dt k k(t) x(t) dt k T n T T n x(t) dt k x(t) k(t)dt k x (t) k(t)dt ck k k k. (-4) The orthogonality of the k, and the ability to interchange integration and summation, have been used to produce (-4). Now, by adding and subtracting a term, write (-4) as n T e x(t) dt n T T n n T k k k k k k k c k k k k x(t) (t)dt x (t) (t)dt c x(t) (t)dt, (-5) n k c k T x(t) k (t)dt which can be organized as (regroup the second line of (-5)) T n T n T n k k k k c k k c k k. (-6) e x(t) dt c x(t) (t)dt x(t) (t)dt Latest Updates at -7

8 Now, select the k to minimize the ISE e n. Note that k only appears in the second term on the right hand side of (-6). Also, this second term is always nonnegative (since the c k > ). Hence, to minimize ISE e n, we should chose the k to zero out the second term. That is, take as the k the values T k x(t) k(t)dt ck, (-7) k n. When these k are used, the minimum integral square error is T n T T n min[e n] x(t) dt x(t) k(t)dt x(t) dt c k k c k k k. (-8) There are infinite sets of basis functions for which the ISE is zero for all x L [, T]. That is, there are infinite-dimensional sets of functions k, k <, for which limit min[e n ] n (-9) and the generalized Fourier series x(t) k k(t) k (-3) holds for all x L [, T]. The k, defined by (-7), are called generalized Fourier coefficients. Sets k, k <, of basis functions for which (-9) holds for all x L [, T] are said to be complete. It is very important to remember that the equality sign in (-3) means equality in the integral square error sense. This is not the same as pointwise equality. As will be seen in the Latest Updates at -8

9 examples which follow, there can be values of t for which x(t) differs from the series expansion. Suppose that an infinite-dimensional complete orthogonal sequence is used so that the ISE approaches zero as n approaches infinity. As a result, from (-8) we get T x(t) dt ck k k, (-3) a result known as Parseval s theorem. Example -3: Consider the set of two orthonormal functions illustrated by Figure -4. Use these functions to compute a best integral-square-error approximation to the signal sin( t), t x(t). (-3), otherwise Using (-7), the coefficients are computed as (t)sin( t)dt sin( t)dt (t)sin( t)dt sin( t)dt. (-33) Thus, the two-term approximation of x(t) is (t) (t) t t Figure -4: Two orthonormal functions. Latest Updates at -9

10 x(t) / x a (t) time -/ x a(t) [ (t) (t)] Figure -5: Sine wave and its two term approximation., (-34) a result that is depicted by Figure -5. Finally, the ISE that results from the approximation is 8 (-35) ISE sin ( t)dt.89 Example -4: Nyquist Low-Pass Sampling Theorem The Nyquist low-pass sampling theorem is an application of generalized Fourier Series. Here, we do not use space L described above; instead, we use the signal space V of all signals x(t), - < t < that are band-limited to W radians/sec. That is, space V consists of all signals x(t), - < t < for which F [x(t)] = for W (verify that this is a valid vector space). We use basis functions of type {sin(x)}/x, and the Fourier coefficients are simply the signal samples x(n/w), - < n <. Latest Updates at -

11 Define (t) Sa( W t n ), - <n<, (-36) n where Sa(x) = {sin(x)}/x. The Fourier transform of n is n F [Sa( Wt n )] exp{ j } W W W, (-37) where Fig. -6 defines. Note that n V for all n. Without proof, we claim that the set of n, - < n < is complete in V; any signal in V can be expanded in a Fourier series that represents the signal as a linear combination of the n. We show that the n are orthogonal. By applying Parseval s theorem we can write n(t) m(t)dt [ n(t)] [ m(t)] d F F. (-38) Now, use (-37) to obtain Sa( Wtn )Sa( Wtm ) dt exp j(nm) d W W W W, (-39) or () -/ / Figure -6: Graph of Latest Updates at -

12 Sa( Wt n )Sa( Wt m ) dt, n m W, (-4), n m and the n are orthogonal as claimed. Now, use the n as a set of basis functions. Let x(t) be a bandlimited signal; that is, assume that X() = F [ x(t)] = for W. Expand x(t) in a Generalized Fourier Series using n = Sa(Wt - n) as the basis functions. The generalized Fourier coefficients are n (t)sa( t n ) dt ( / ) x W W (-4) Now, let X() = F [x(t)], and use Parseval s theorem to write n (t)sa( t n ) dt ( / ) x W W n ( ) exp j Rect ( )d ( / ) X W W W W n ( )exp j d (since ( ) for ) X X W W (-4) n = x W Hence, bandlimited x(t) can be reconstructed from its samples by the generalized Fourier series n x(t) x Sa( Wt n ) n W. (-43) This result is the well-known Nyquist low-pass sampling theorem, and it is simply an application Latest Updates at -

13 of the theory of generalized Fourier series. Exponential Fourier Series Let x(t) be defined on the interval [, T]. For the exponential Fourier Series, we use the complete orthogonal set k = exp[jk t], where = /T. For these basis functions, we have c e dt T k T jk t, (-44) so that x(t) jkt ke, (-45) k where T jkt k x(t)e dt T, (-46) and = /T. Series (-45) represents x(t) on the interval [, T]; if x(t) is T-periodic, the series represents x(t) on - < t <. If x(t) is continuous at t [, T], the series (-45) converges to the value x(t ). If x(t) has a jump discontinuity at t [, T], then series (-45) converges to the midpoint x(t ) x(t ). (-47) In general, the series (-45) will not represent x(t) outside of the interval [, T]. For the special case where x(t) is T-periodic, the series represents x(t) for all t. The double-sided amplitude spectrum of T-periodic x(t) is a plot of k versus k. The Latest Updates at -3

14 double-sided phase spectrum of T-periodic x(t) is a plot of k versus k. Usually, phase is defined and plotted in a modulo- manner. That is, phase k is defined so that k, k. (-48), k k There exists commonly-used terminology for the T-periodic case. is the DC component. Frequency is the fundamental frequency. For k, k is the frequency of the k th harmonic. Example -5: Half-Wave Rectified Sine Wave Consider the signal x(t) Asin( t), t T/, (-49), T / t T where = /T, and x(t) = x(t + T). The Fourier series coefficients are T/ jk t k T Asin( t)e dt (-5) For k =, this can be computed as j t T/ t T/A jt A e A e dt t j T j jt j 4 (-5) t For k, k, we can integrate by parts, or use an integral table, to obtain Latest Updates at -4

15 T/ jkt jkt A e tt/ k Asin( t)e dt jk sin( t) cos( t) T T t (jk ) jkt / A e A T T (jk) (jk) jk sin( T / ) cos( T / ). (-5) But, T =, so that (-5) becomes jk e A ( ) k A jksin( ) cos( ) A (k ) (k ) (k ) k A, k even (k ),, k odd, k for k, k. Plots of the double-sided amplitude and phase spectrum are depicted by Figure -7. Symmetry Properties of Exponential Fourier Series Coefficients Indexed on integer k, let k denote the exponential Fourier series coefficients. If the original function x(t) is real-valued, then it is easy to see that k k (-53) for all k. Hence, this fact implies the symmetry properties k k k k. (-54) That is, the magnitude is an even function of index k while the phase is an odd function. Also, it Latest Updates at -5

16 Amplitude Spectrum k A/ A/ A/ A/ A/ A/ A/ o o o o o o o o n o Mod- Phase Spectrum k / o o o o o o o o n o / Figure -7: Double-sided amplitude and mod- phase spectrum of a half-wave rectified sinusoid. is easily shown that Re[ k ] is an even function of k, and Im[ k ] is an odd function. Parseval s Theorem for Exponential Fourier Series Let x(t) be a T-periodic signal. For the exponential Fourier series, Parseval s theorem says that average power P can be expressed as T x(t) dt k T k P (-55) Think of Parseval s theorem as giving two ways (using time and frequency domain principles) to compute the per-ohm average power of a periodic signal. Single-Sided Amplitude and Phase Spectra for Real-Valued, Periodic Signals Suppose x(t) is a real-valued, T-periodic signal. The exponential Fourier series can be written as Latest Updates at -6

17 jkt jkt k k k k x(t) e Re e j jk t j( k t ) Re k e e k Re e k k k k (-56) cos(k t ) k k k A plot of at the origin and k versus k, k >, is known as the single-sided amplitude spectrum. A plot of k versus k, k >, is known as the single-sided phase spectrum. The Fourier Transform The Fourier series cannot represent an aperiodic function x(t) on the entire time line - < t <. In what follows, we seek a frequency domain representation for aperiodic signals. This will lead to the Fourier transform of x(t) which we derive as the formal limit of a Fourier series. Let x(t) be defined on the entire time line - < t <. Use an exponential Fourier series to expand x(t) on the finite interval [-T/, T/]; this results in jk t k x(t) e, T/ t T/ k T k T/ jkt x(t)e T/ dt, (-57) where = /T. Define T/ jkt X(k ) T k x(t)e dt T/ (-58) so that x(t), -T/ t T/, can be represented as Latest Updates at -7

18 jkt x(t) X(k )e, T / t T/ T k. (-59) Now, consider the formal limit of (-59) as T, k (a new frequency variable), and (/T) d. In the limit (-59) and (-58) become jt x(t) X( )e d (-6) and j t X( ) x(t)e dt, (-6) respectively. Transform (-6) is the Fourier transform, and (-6) is the inverse Fourier transform. Double-Sided Amplitude and Phase Spectra of Fourier Transforms Often, plots are used to depict a Fourier transform. First, we write the transform in terms of its magnitude and phase as X( ) M( )exp[j ], (-6) where M() = X() = A dc () + M ac (), and () = X(). Here, we use A dc to denote the amplitude of a possible discrete DC component in the wave form, and M ac denotes the AC spectra (i.e., M ac contains no impulse at = ). A plot of M(), - < <, is known as a double-sided amplitude spectrum, while a plot of (), - < <, is known as a double-sided phase spectrum. Usually, we plot phase in a modulo- manner; for <, define phase so that - <, and for >, define phase so that - <. Latest Updates at -8

19 Example -6: Double-Sided Spectra of a Pulse Consider the rectangular pulse depicted by Figure -8. First, assume that t = so that the pulse is symmetrical with respect to the origin. For the case t =, we can compute j t / A sin( / ) X( ) x(t)e dt A cos( t)dt sin( / ) A / ASa /, (-63) where Sa(x) sin(x)/x, a standard function in signal processing and communication systems. Now, use the time-shifting theorem to compute the Fourier transform of A[(t-t )/] as j t. (-64) X( ) A e Sa / For the case t = /, the double-sided amplitude and phase spectra is depicted by Figure -9 (phase is plotted in a modulo- manner). For t = /6, Figure - depicts the spectra. Note that the delay value t has a significant effect on the phase function () (what would the phase plot look like for t = /4? For t = /8?). Single-Sided Amplitude and Phase Spectra of the Fourier Transform If x(t) is real-valued, it is easy to show that M() = X() and R() = Re[X()] (the real part) are even functions of Likewise, () = X() and I() = Im[X()] (the imaginary part) are odd functions. Write X() = M()exp[j()] = A dc () + M ac ()exp[j()]. The A[(t-t )/] A t Figure -8: A[(t-t )/], a rectangular pulse of width which is centered at t. t Latest Updates at -9

20 Absolute Magnitude X t = A Absolute Magnitude X() t = /6 (radians/second) (radians/second) X t = X() Phase in Radians Phase in Radians - - t = /6 - (radians/second) Figure -9: Double-sided magnitude and phase spectra for the case t = /. (radians/second) Figure -: Double-sided magnitude and mod- phase spectra for the case t = /6. quantity A dc is the DC component (which may be zero) in x(t), and M ac () contains no impulse at =. If x(t) is real-valued, we can write jt j ( ) jt x(t) X( )e d [A dc( ) M ac( )e ]e d Adc M ac( )cos{ t ( )}d. (-65) For, a plot of A dc () + M ac () is known as the single-sided amplitude spectrum of the real-valued signal x(t). Likewise, for, a plot of () is known as the single-sided phase spectrum. No negative frequencies are displayed on the plots. For >, note the factor of two difference between the single- and double-sided amplitude spectra (both amplitude plots display the same discrete DC component). Latest Updates at -

21 Parseval s Theorem for Fourier Transforms of Energy Signals The energy in an energy signal can be expressed as jt jt E x(t) dt x (t) X( )e d dt X( ) x (t)e dt d,(-66) X( )X ( )d which is the same as E x(t) dt X( ) d (-67) Equation (-67) is Parseval s Theorem for Fourier transforms. Think of this result as providing two methods for calculating the energy in an energy signal. The quantity G( ) X( ) (-68) is called the energy density spectrum of energy signal x(t); it has units of Joules/Hz. Convolution The convolution of signals f(t) and g(t) is a new function of time. It is defined by f(t) g(t) f( )g(t )d. (-69) A simple change of variable produces f(t) g(t) f( )g(t )d f(t )g( )d. (-7) Latest Updates at -

22 Hence, you can fold and shift f or g in the convolution integral. A close inspection of (-69) reveals that convolution involves the operations ) fold g() to produce g(-), ) shift g(-) to form g(t-), and 3) integrate the product f()g(t-) over the range - < <. Example -7: Use formula (-69) to find where - g f = g(t - )f ( )d f() g() CASE : t < t- g(t-) t f() -axis No overlap fg = for t < CASE : < t < g(t-) f() -axis t Overlap region from to t t ( [t ])d (t )d t tt /, < t < Latest Updates at -

23 CASE 3: < t < f() g(t-) ( [t ])d t t /, < t < t -axis t- Overlap region from t- to CASE 4: t > f() g(t-) No overlap fg =, t > t- t -axis No Overlap The convolution of f and g is f g, t tt /, t tt /, t t t + t / fg The result of the convolution is depicted by Figure -... t - t / Figure -: Result of the convolution of f and g. t Latest Updates at -3

24 Superposition Theorem If x (t) X () and x (t) X () then, for any constants and, we have x (t) x (t) X ( ) X ( ), (-7) a result that follows from the fact that integration is a linear operation. Time Shifting Theorem If x(t) X(), then for any shift t, we have j t x t t e X( ). (-7) Proof: F j x(tt ) x(tt )e dt x( )e de x( )e d. (-73) j t j ( t ) j t j t e X( ) Time Scaling Theorem If x(t) X(), then for any non-zero, real-valued constant, we have xt X( / ). (-74) Proof: First assume that > and write j t j( / ) Fx( t) x( t) e dt x( ) e d X( / ). (-75) Latest Updates at -4

25 Next, consider the case < and write jt j( / ) j( / ) Fx( t) x( t) e dt x( ) e d x( ) e d. (-76) X( / ) Now, recall the definition,. (-77), Combine the last three equations to obtain (-74). Symmetry Theorem The Fourier transform is symmetrical. That is, x(t) and X() are transform pairs, then we have the two pairs x(t) X( ) X(t) x( ) (-78) Proof: Simply interchange t and (after all, they are just algebraic variables) in jt x(t) X( ) e d to obtain (after multiplication by ) jt x( ) X(t)e dt. Latest Updates at -5

26 Time Domain f(t) = rect(t/ ) F() = Sa( ) Frequency Domain t (Rad/Sec) Figure -: Fourier transform found in almost all transform tables. Now, in this last equation, replace by - to obtain j t x( ) X(t)e dt. This last equation is the desired result X(t) x(-), and this establishes (-78). Example -8: Suppose we need the Fourier transform of f(t) = Sa( t), where Sa(t) = sin(t)/t. We cannot apply the definition directly; the integral is too hard to integrate. We cannot find the time-domain function f(t) = Sa( t) in our table of Fourier transforms. However, in our table, we do see the transform pair F[rect(t/ )] = Sa( ), as depicted by Figure -. Application of the Symmetry Theorem allows us to flip this pair and obtain the desired results. So, we conclude that the desired transform is F[ Sa( t)] = rect(/ ), as shown by Figure -3. Frequency Shifting Theorem If x(t) X() then for any frequency shift we have j t (-79) x(t)e X( ) Proof: Observe that Latest Updates at -6

27 Time Domain Frequency Domain F(t) = Sa(t ) f() = rect (/ ) t (Sec) (Rad/Sec) - Figure -3: Transform pair obtained by applying the symmetry theorem to the pair depicted by Figure -. j t j t j t j( )t F x(t)e {x(t)e }e dt x(t)e dt. (-8) This equation implies j t, (-8) x(t)e X( ) the desired result. Differentiation Theorem If x(t) X() then n d x dt n n (j ) X( ) (-8) Proof n n n jt jt n jt n n n d x d d X( )e d X( ) e d {(j ) X( )}e d dt dt dt - n F (j ) X( ) Latest Updates at -7

28 This last result establishes the desired results n d x dt n n (j ) X( ). Convolution Theorem That is, Convolution in the time domain gives rise to multiplication in the frequency domain.. x x x ( )x (t )d x (t )x ( )d X ( )X ( ) Proof j t j t F [x x ] e x ( )x (t )d dt x ( ) e x (t )dt d j x()x ( )e d X( )X ( ) Integration Theorem If x(t) X(), then t X( ) x( )d X() ( ). j Proof: Note that the left-hand side of the previous equation can be expressed as t x( )d U(t) x(t), Latest Updates at -8

29 where U(t) is the unit step function. Use the Convolution Theorem, and take the Fourier transform of this last equation to obtain t F x( )d U(t) x(t) U(t) x(t) ( ) X( ) j F F F X( ) ( )X(). j (-83) Multiplication Theorem That is, Multiplication in the time domain corresponds to convolution in the frequency domain. x (t)x (t) X ( ) X ( ) (-84) Proof: Similar to the Convolution Theorem given above. Fourier Transform of Periodic Signals Let x(t) be T-periodic with exponential Fourier series expansion x(t) jnt ne, n where = /T. On a term-by-term basis, take the Fourier transform of the series to obtain F [x(t)] ( n ), (-85) n n a result depicted by Fig -4. Latest Updates at -9

30 F [x] Cross Correlation and Autocorrelation of Energy Signals 3 Figure -4: Fourier Transform of Periodic Signal Let x(t) and y(t) be complex-valued energy signals. Their cross correlation is defined as R xy( ) x (t)y(t )dt. (-86) Crosscorrelation is non-comutative; that is, R xy () R yx (). However, R xy( ) R yx ( ) (can you show this result?). The autocorrelation of complex-valued signal x(t) is defined as R x ( ) x (t)x(t )dt (-87) Note that x x R ( ) R ( ) and that R () x (t)x(t)dt x(t) dt Signal Energy x. (-88) An upper bound on R xy () is given by R xy( ) R x () R y(). (-89) We prove this for the case of real-valued signals x and y (the case where x and y are complexvalued is slightly more complicated). Latest Updates at -3

31 [x(u) y(u )] du x(u) du y(u ) du x(u)y(u )du R () R () R ( ) x y xy (-9) Hence (-89) follows as claimed. If in this last result we use x = y then we have R x () R x () for all ; that is, the maximum autocorrelation occurs at the origin. Example -9: Calculate the cross correlation x(t) R xy( ) x (t)y(t )dt for the two real-valued signals y(t) Case : >, no overlap t t y(t+) x(t) - -+ t R xy( ), > Case : < < y(t+) x(t) - -+ overlap from to - + Case 3: - < < t xy R ( ) ( t) dt [ ], < < x(t) y(t+) xy R ( ) ( t) dt [ ], < < t - Latest overlap Updates from at - to -3

32 Case 4: < - x(t) y(t+) R xy( ), < - - t Summary: R xy( ),, > [ ], < < [ ], < <, Relationship between Crosscorrelation and Convolution Find the relationship between R xy( ) x (t)y(t )dt and xy x(t)y( t)dt, (-9) crosscorrelation and convolution, respectively. Use the change of variable v = t + in R xy to obtain R ( ) x (v )y(v)dv x ( [ v])y(v)dv xy, (-9) which is just the convolution of x*(-t) and y(t). Hence, R xy () can be computed by convolving x*(-t) and y(t); that is, Latest Updates at -3

33 R xy( ) x*(- ) y( ). (-93) Fourier Transform of Crosscorrelation - Energy Density Spectrum Let x(t) X() and y(t) Y() be Fourier transform pairs. From (-93) and the Convolution Theorem, we obtain F R xy( ) F x*(-t) y(t) X ( )Y( ). (-94) For autocorrelation we have G( ) F R ( ) F x*(-t) x(t) X ( )X( ) X( ), (-95) x the energy density spectrum of energy signal x(t). Hence, the Fourier transform of the autocorrelation is the energy density spectrum. Cross Correlation and Autocorrelation of Power Signals First, we define a time averaging operation as z(t) limit z(t)dt T T T. (-96) T For example, x (t) is the average power in power signal x(t). Let v(t) and w(t) be power signals. The average v*(t)w(t) is called the scalar product (or dot product) of v(t) and w(t). The scalar product may be complex valued; it serves as a measure of similarity between the two signals. We define the cross correlation of power signals v and w as R ( ) v (t)w(t ) v (t )w(t). (-97) vw Latest Updates at -33

34 Note that R ( ) v (t)w(t ) v(t)w (t ) v(t )w (t) w (t)v(t ), (-98) vw and this implies the symmetry condition vw wv R ( ) R ( ), (-99) a formula identical to that previously obtained for energy signals. The autocorrelation of power signal v(t) is defined as R ( ) v (t)v(t ) v (t )v(t). (-) v It is possible to show: ) R v () R v () = Avg Pwr {v(t)}, so R v () has its maximum at = (this follows from (-)). ) Since R v( ) R v( ), if v(t) is real-valued, then Rv () will be real-valued and even so that R v () = R v (-). 3) If v(t) is T-periodic, then R v () is T-periodic. 4) If v(t) does not contain discrete frequency sinusoidal components then v limit R ( ) v(t) DC Power in v(t). (-) Given power signals x(t) and y(t), we can show Latest Updates at -34

35 R xy( ) R x () R y(). (-) This result can be proved in a manner that is identical to that used to establish (-89) for the energy signal case. Equation (-) establishes an upper bound on the cross correlation of two power signals. Note that R xy () may have a maximum for. The fact that R vw () and R v () always exists for all follows from Schwarz s Inequality, a result that is proved below. Schwarz s Inequality states that v(t)w*(t) v(t) w(t) P P (-3) v w for any power signals v and w. (P v and P w are the finite powers of v and w, respective.) Now, the right-hand side of this last result is the product of the average powers of the two signals, and it is finite. Apply (-3) to R vw () and conclude that R vw () P v P w. Hence, the magnitude R vw () is finite for all, and the cross correlation exists for all time shifts. Schwarz s Inequality Let P v and P w denote the average powers of power signals v(t) and w(t), respectively. Then we have v(t)w*(t) v(t) w(t). (-4) P v P w (w*(t) can be replaced with w(t) and the inequality still holds). Proof: Define the power signal z(t) v(t) w(t), (-5) Latest Updates at -35

36 where is a to-be-defined complex number. The average power in z(t) is Pz z(t)z (t) {v(t) w(t)}{v (t) w (t)} Pv Re v(t)w (t) Pw, (-6) where we have used v(t)w (t) = v (t)w(t). Now, set = v(t)w (t)p w and obtain v(t)w (t) v(t)w (t) Pz Pv Re P w Pw Pw (-7) v(t)w (t) Pv Pw from which the desired results (-4) follows. Equality in (-4) holds if, and only if, v(t) is proportional to w(t); that is, equality in (-4) holds iff v(t) = kw(t) for some constant k. Uncorrelated Signals Two signals v(t) and w(t) are said to be uncorrelated if R vw () = for all. Let v(t) and w(t) be uncorrelated. Note that R ( ) {v (t) w (t)}{v(t ) w(t )} vw v (t)v(t ) v (t)w(t ) w (t)v(t ) w (t)w(t ). (-8) However, the cross terms are zero so that (-8) becomes Latest Updates at -36

37 R vw( ) R v( ) R w( ). (-9) Hence, for two uncorrelated signal, the autocorrelation of the sum is the sum of the autocorrelation. Setting =, we see that the power of the sum is the sum of the powers for uncorrelated signals (this is not generally true for correlated signals). Crosscorrelation of T-periodic Power Signals in Terms of Fourier Series If x(t) and y(t) are periodic with period T, then cross correlation R xy () is periodic with period T. Furthermore, to calculate the cross correlation, you only need to integrate over one period of the waveforms. That is, for T -periodic x(t) and y(t), we can compute T R xy( ) x (t)y(t )dt T. (-) For the periodic case just considered, we can express the Fourier series of R xy () in terms of the Fourier series of x and y. Express T -periodic x and y as jkt x(t) k e k jk t y(t) e, k k (-) where = /T is the fundamental frequency of the signals. Then, we can write T T jnt jmt jm xy n e m e e T T n m R ( ) x (t)y(t )dt dt (-) jm T j(mn) t nm T nm e e dt. Latest Updates at -37

38 Now, the complex exponential functions are orthogonal so that T T j(mn) t e dt, n m (-3), n m, and this fact leads to the simple formula xy n n jn (-4) n R ( ) e for the T -periodic cross correlation. Similarly, T -periodic x(t) has a T -periodic autocorrelation given by R ( ) x jn n e. (-5) n Power Spectral Density We are interested in defining the power spectrum S() of power signal x(t). The power spectrum should be a real-valued, nonnegative function. It should be an even function of if x(t) is real-valued. Furthermore, the total average power should be obtained by integrating S()/ over all frequency, that is, the total average per-ohm power should be obtained from Pavg limit x(t) dt ( )d (watts) T T T T S. (-6) From (-6), it is easily seen that the units associated with S() are watts/hz. Example -: Consider a T-periodic power signal x(t). In a Fourier series, expand x(t) as Latest Updates at -38

39 x(t) jnt ne. (-7) n The power spectrum of the T-periodic signal is n n S ( ) ( n ). (-8) This follows since ( )d n ( n )d n P avg n n S (-9) by Parseval s Theorem. We seek a general method to represent/compute the power spectrum. Towards this end, consider windowing power signal x(t) to form x T(t) x(t)rect(t /T), (-) where rect(t/t) =, T / t T / =, otherwise (-) is a T-long time window centered at the origin. Windowed signal x T is an energy signal, and its Fourier transform is denoted by XT( ) F x(t)rect(t/t). (-) Latest Updates at -39

40 By Parseval s theorem for Fourier transforms, we have x (t) dt ( ) d T/ T/ T T X (-3) for all T. Hence, the average power is given by X ( ) limit x (t) dt limit X ( ) d limit d. (-4) T T T T/ T T/ T T T T T Hence, from (-6) and (-4) we see that XT T ( ) ( )d limit d S. (-5) T As part of our definition of S, we require that (-5) hold over arbitrary frequency bands (frequency intervals). This requirement leads to XT T ( ) ( )d limit d S (-6) T for all values of. Basically, we get the power that is contained in the (, ) frequency band if we integrate S()/ over this band. Now, we can write a general formula for the power spectrum. With respect to frequency, differentiate (-6) to obtain T XT( ) S ( ) limit, (-7) T Latest Updates at -4

41 an important formula for the power spectrum. Equation (-7) suggests a practical way to approximate the power spectrum of a power signal. First, a data record of length T is recorded. Next, the Fourier transform X T () of the time record x T (t) is computed/approximated (often, by an FFT). Finally, the power spectrum is approximated by X T () /T, a result that may be plotted as a function of frequency. Example -: Use (-7) with a large value of T to approximate the power spectrum of power signal Aexp(j t). How does the approximation behave as T approaches infinity? First, consider the windowed signal x T = Aexp(j t)rect(t/t) with transform XT( ) F Aexp(jt)rect(t/T) AT Sa ( )T / (-8) so that for large, finite T we have T ( ) X S ( ) = A T Sa ( )T/ T (-9) S( ) A T Sa[( T ) ] A T Width 4/T Figure -5: Approximation to power spectrum. approximation approaches A (- ) as T. The Latest Updates at -4

42 as our approximation to the power spectrum. This approximation is illustrated by Figure -5; note that the base width is proportional to 4/T and the height is proportional to A T. In the limit as T, this last equation produces T T ( ) X S( ) limit = A limit T Sa ( )T / T T T A limit Sa ( )T/ A ( ) T (-3) as the power spectrum. System Input - Output Cross Correlation and Output Autocorrelation Let x(t) and y(t) denote the input and output, respectively, of a linear, time-invariant system having h(t) as its impulse response. We show that R xy (t) = h(t)r x (t) for the case where x and y are energy signals; the result is true for power signals as well. Let x and y be energy signals and observe R xy( ) x (t)y(t )dt x (t) h(u)x(t u)du dt h(u) x (t)x(t u)dt du h(u)r x( u)du (-3) h( ) R ( ). x Hence, we have R xy () = h()r x () as claimed. As stated previously, this result holds for power signals as well as energy signals. It is possible to compute the output autocorrelation from knowledge of the input autocorrelation and the system s impulse response. To see how this is done, consider Latest Updates at -4

43 R y( ) y (t)y(t )dt h (u)x (t u)du y(t )dt h (u) x (tu)y(t )dt du. (-3) In this result replace the t variable with v t - u to obtain R y( ) h (u) x (v)y(v u )dv du h(u)r xy(u )du h( u)r xy( u)du (-33) h( ) R (), xy the convolution of h * (-) and R XY (). Now, substitute (-3) into (-33) and obtain y xy x R() R () h( ) h() R () h( ) h( ) h ( ) R x ( ), (-34) so that the autocorrelation of the output can be expressed as the autocorrelation of the input convolved with h()h*(-). Power Spectrum as Fourier Transform of Autocorrelation - The Wiener-Khinchine Theorem Recall that the power spectrum of power signal x(t) is T X T( ) S ( ) limit. (-35) T Take the inverse Fourier transform of S to obtain Latest Updates at -43

44 F - X T( ) j S( ) T T T/ j t T/ jt j T/ T/ limit x (t )e dt x(t )e dt e d T T/ T/ j (t t ) T/ T/ limit x (t ) x(t ) e d dt dt T limit e d T T. (-36) However, from Fourier transform theory we know that j (tt ) e d (tt ). (-37) Now, use (-37) in (-36) to obtain F - S( ) limit x (t ) x(t ) (t t )dt dt T T T/ T/ T/ T/ T/ limit x (t)x(t )dt x (t)x(t ) T T/ T. R ( ) x This is the well-known, and very useful, Wiener-Khinchine Theorem: the Fourier transform of the autocorrelation is the power spectrum density. Symbolically, we write R x ( ) ( ) S. (-38) Power Signals in Systems: Input/Output of Power Spectrums Suppose power signal x(t) is input to a linear, time-invariant system to produce y(t), the output power signal. How can we find the output power spectrum in terms of the input power Latest Updates at -44

45 spectrum? By the Wiener-Kinchine theorem, we can write the power spectrum of output y(t) as S. (-39) y( ) F R y( ) Use the fact that R y () = [h(t)h * (-t)]r x () to write S h(t) h ( t) h(t) h ( t) ( ) F R ( ) F R ( ) F F R ( ). (-4) y y x x However, F [h * (-t)] = H * (j) so that S y x ( ) F h(t) h ( t) F R x ( ) H (j )H(j ) Sx( ), (-4) H(j ) S ( ) an important result that is used in many applications in signal processing, radar, etc (see Fig. - 6). Periodic Signals Let x(t) be T-periodic with Fourier series expansion x(t) jkt k e, (-4) k x(t) h(t) y(t) Input Power Spectrum S x () Output Power Spectrum S y () Figure -6: Linear system. Output spectrum is related to input spectrum by S y () = H(j) S x (). Latest Updates at -45

46 where = /T. Recall from (-5) that the autocorrelation of x is R ( ) x jn n e. (-43) n From the Wiener-Kinchine Theorem we have the power spectrum as x n n n jn n F S( ) F R ( ) e ( n ). (-44) Note that all of the power is concentrated at DC and the various discrete frequency components in the signal. Signals and Systems A system is a mapping between an input x(t) and an output y(t). Notice that our system, depicted by Fig. -7, has a single input and a single output. It is possible to generalize this concept to systems with multiple inputs and outputs. Linearity The system is said to be linear if it obeys superposition. Let x and x be arbitrary inputs; assume that x and x produce outputs y and y, respectively. Then the system is said to linear if, for any constants and, the input x + x produces the output y + y. Let h(t,) denote the system s response at time t to an function applied at time, an input/output relationship depicted by Fig. -8. Function h(t,) is called the system s impulse response. It is well-known that the system s output y(t) is given by x(t) L[ ] y(t) = L[x(t)] Figure -7: A system with input x(t) and output y(t). Latest Updates at -46

47 Input System Output h(t;) (t-) L[ ] t Figure -8: An impulse applied at time produces h(t,) at time t. t y(t) h(t, )x( )d, t. (-4) As discussed below, this input-output (I/O) relationship can take on various forms depending on the system properties of time invariance and causality. Time Invariance Let x(t) and y(t) be an arbitrary input/output pair. The system is said to be time invariant if input x(t - t ) produces output y(t - t ) for any time shift t. This idea is conveyed by Figure - 9. For a time-invariant system, the output at time t to an input applied at time depends on the time difference t - and not on the absolute value of t or. For this reason, h(t, ) = h(t - ), and IF: x(t) y(t) L[ ] THEN: x(t-t ) t y(t-t ) t L[ ] t t t t Figure -9: Time-invariance: if y(t) = L[x(t)] are any input-output pair, then y(t-t ) = L[x(t-t )] for any t. Latest Updates at -47

48 the input-output relationship (-4) becomes y(t) h(t )x( )d, t, (-4) since the impulse response depends on the time difference and not absolute t or. Equation (-4) states that output y(t) is the convolution of input x(t) and impulse response h(t). Causality Let x (t) and x (t) be arbitrary inputs which produce outputs y (t) and y (t), respectively. The system is said to be causal if x (t) = x (t), t < t, for any t, implies that y (t) = y (t) for t < t. Note that x (t), x (t) and t are not system dependent (they are arbitrary except for the requirement that x (t) = x (t), t < t for some t ). Basically, a causal system cannot respond before it is excited; its output y at time t does not depend on input x at time greater than t. One can show that a linear, time-invariant system is causal if, and only if, its impulse response h(t) vanishes for t <. That is, a necessary and sufficient condition for causality is h(t), t. (-4) For linear, time-invariant causal systems, relationship (-4) becomes t y(t) h(t )x( )d, t, (-43) where it is assumed that input x(t) starts (was initially applied) in the infinite past. Bounded Input - Bounded Output Stability (BIBO Stability) A linear system is said to be bounded input - bounded output stable (BIBO stable) if bounded inputs always produce bounded outputs. It can be shown that a linear, time-invariant system is BIBO stable if, and only if, Latest Updates at -48

49 h(t) dt. (-44) That is, the system is BIBO stable if, and only if, h(t) is absolutely integrable. Transfer Function Linear, time-invariant systems can be described in the frequency domain by using the Fourier transform. Take the Fourier transform of (-4) to obtain Y( ) H( )X( ), (-45) where Y, H and X are Fourier transforms of y, h and x, respectively. H() is called the transfer function of the system. If h(t) is real-valued, then Re[H()] is even and Im[H()] is odd (so that H() is even and H() is odd). A plot of H() is called the double-sided amplitude response, and a plot of H() is called the double-sided phase response. Example -: Consider the circuit depicted by Figure -. A relationship between input x and output y is given by dy dy RC y x y x, (-46) dt dt where RC is the system time constant. From elementary ordinary differential equation t/ theory, use the integrating factor (t) = e and express the complete solution of (-46) as R + + x(t) C y(t) - - Figure -: A simple first-order system Latest Updates at -49

50 t/ t / y(t) e e x( )d, C (-47) where C is an arbitrary constant of integration (note the indefinite integral within the bracket). Given initial condition y(), the solution to the initial value problem (IVP) is t (t )/ t/ y(t) e x( )de y(), t.. (-48) Instead of starting at t =, suppose we are given the initial condition y(t ) and asked to find y(t), t t >. Clearly, we have y(t) e x( )de y(t ), t t t (t )/ (tt )/ t, (-49) where y(t ) is the initial voltage across the capacitor at t = t. Assume that all initial conditions are zero. From (-48), for t, the output is given by t y(t) h(t )x( )d, t, (-5) where t/ h(t) e U(t) (-5) is the circuit impulse response. The transfer function of this system is given by Latest Updates at -5

51 H( ) F [h(t)] j /, (-5) where = /RC is the 3dB cut off frequency of the filter. Periodic Response to Periodic Input A T-periodic response can be obtained from a linear, time-invariant system that is driven by a T-periodic forcing function. Let the input x(t) be represented as n n jnt x(t) c e. (-53) Let the system impulse response be h(t) so that the output can be computed as jn (t ) jn jnt y(t) cne h( )d e h( )d cne n n jnt H(n )cne n, (-54) where H() = F [h(t)], the Fourier transform of the impulse response (i.e., the transfer function). Note that c n H(n ), - < n <, are the Fourier series coefficients of the periodic output. For the simple case where x(t) = Acos( t), the output is A j t j t j{ t H( )} y(t) H( )e H( )e A Re H( ) e. (-55) A H( ) cos{ th( )} Piecewise Description of Periodic Response Often, with simple first- and second-order systems, you can piece together a periodic Latest Updates at -5

52 solution by using simple techniques. Example -3: Consider the RC network depicted by Figure - with R = Ohm and C = Farad. Suppose input x(t) is a T-periodic square wave that toggles between. When x =, the capacitor is charging up with a time constant = RC = second. Starting from at t =, the capacitor voltage increases exponentially, finally reaching at t = T/. When x = -, the capacitor is discharging; the capacitor voltage exponentially decreases, reaching at t = T. Figure - depicts one period of the input and anticipated response. The values and must be computed. From (-48) with = RC =, y() = and y(t/) =, we can write T/ (T/ ) T/ y(t / ) e ( )d e. (-56) Application of (-49), with x(t) = - over T/ t T, produces T (T ) T/ T/ y(t) e ( )de. (-57) In general, (-56) and (-57) must be solved for and. However, the input is a symmetrical perfect square wave so = - and (-56) produces T/ T/ ed e, a result that can be solved for T/ e T/ e. (-58) Latest Updates at -5

53 Input T/ T Output Time (Seconds) Figure -: Square wave input shown as solid-line graph. RC circuit output shown as dashed line graph. Values of and are dependent on R, C and T, and it must be computed. Over the range t T/, the output can be expressed as t (t ) t t y(t) e de ( )e, t T /. (-59) On the second-half of the period, the output can be expressed as t (t ) (tt/) (tt/) T/ y(t) e ( )de ( )e, T/ t T. (-6) In (-59) and (-6), use given by (-58). The T-periodic output is simply a periodic extension of the waveform described by (-59) and (-6). Figures - and -3 depict one period of the output for T = 4 and T =, respectively. The simple techniques that were used here for the RC circuit can be applied to a wide number of first- and second-order systems subjected to a wide variety of periodic inputs. Latest Updates at -53

54 Output Volts Response When T = 4 Seconds 3 4 Time in Seconds Figure -: One period of the output for the case T = 4 seconds. Output Volts Response When T = Seconds Time in Seconds Figure -3: One period of the output for the case T = seconds. Ideal Low-Pass Filter An ideal low-pass filter (LPF) has a transfer function given by j t H lp( ) Ae, c c,, otherwise (-6) where A >, t > and bandwidth c are known. Figure -4 depicts the magnitude and phase response of an ideal LPF. Within the pass band, the amplitude is constant and the phase is linear in frequency. The impulse response of the ideal LPF is A H lp (j) c c H lp (j) -t Figure -4: Magnitude and phase response of an ideal low-pass filter. A and t are known quantities. Latest Updates at -54

55 A h c lp(t) Sa c (tt ) A c t t (Sec) Figure -5: Impulse response of an ideal low-pass filter. - Ac sin c(tt ) A h c lp(t) H lp( ) Sa{ c (tt )} c(t t ) F, (-6) where Sa(x) = sin(x)/x. Figure -5 depicts the general form of (-6). Note that the ideal LPF is non-causal, which implies that real-time, ideal low-pass filters cannot be implemented. Ideal High-Pass Filter An ideal high-pass filter (LPF) has a transfer function given by j t H hp( ) Ae, c,, c (-63) where A >, t > and c are known. Figure -6 depicts the magnitude and phase response of an ideal HPF. Note that hp j t H ( ) Ae H ( ), (-64) lp so that the impulse response of an ideal high-pass filter is given by Latest Updates at -55

56 A H hp (j) c c H hp (j) Figure -6: Transfer function of an ideal high-pass filter. -t - A h c hp(t) H hp( ) A (tt ) Sa{ c (tt )} F, (-65) as shown by Figure -7. Ideal Band-Pass Filter A band-pass filter passes signals contained in a band centered at some frequency. An ideal band-pass filter (BPF) has a transfer function given by A h c hp(t) A (t t ) Sa{ c (t t )} A(t-t ) t (Sec) t A c Figure -7: Impulse response of an ideal high-pass filter. Latest Updates at -56

57 A H bp (j) c H bp (j) Figure -8: Magnitude and phase response of an ideal band-pass filter. -t j t H bp( ) Ae, c and c, otherwise, (-66) where A >, t >, > and c > are known. Figure -8 depicts the magnitude and phase response of an ideal BPF. Note that j t j t bp lp lp H ( ) H ( )e H ( )e, (-67) so that the impulse response of an ideal high-pass filter is given by j (tt ) j (tt ) j (tt ) j (tt ) e e bp lp lp lp h (t) h (t)e h (t)e h (t) h lp(t)cos (t t ), (-68) see Figure -9. Latest Updates at -57

58 A c h bp(t) h lp(t)cos (t t ) t t (Sec) Figure -9: Impulse response of an ideal band-pass filter. Butterworth Low-Pass Filters We describe the commonly-used n th -order Butterworth filter. This is accomplished by stating the filter magnitude response, filter pole locations, and filter transfer function. The squared magnitude function for an n th -order Butterworth low-pass filter is H(j ) H(j )H (j ), (-69) n (j / j ) c where constant c is the 3dB cut-off frequency. Magnitude H(j) is depicted by Figure -3. It is easy to show that the first n- derivatives of H(j) at = are equal to zero. For this reason, we say that the Butterworth response is maximally flat at =. Furthermore, the derivative of the magnitude response is always negative for positive, the magnitude response is monotonically decreasing with increasing. We require h(t) to be real-valued and causal. This requirement leads to jt jt, (-7) H (j ) h(t)e dt h(t)e dt H( j ) Latest Updates at -58

59 Magnitude H() n = n = 4 n = 6 so that / c Figure -3: Magnitude response of an ideal n th -order Butterworth filter. H(j )H( j ). (-7) n (j / j ) c Note that there is no real value of for which H() =. That is, H(j) has no poles on the jaxis of the complex s-plane. We require the filter to be causal and stable. Causality requires h(t) =, t <. Stability requires the causal impulse response to satisfy h(t) dt. (-7) Causality and stability requires that the time-invariant filter have all of its poles in the left-half of the complex s-plane (no j-axis, or right-half-plane poles). The region of convergence for H(s) is of the form Re(s) > for some <, see Figure -3. Hence, H(s) can be obtained from H(j), the Fourier transform of the impulse response, by replacing j with s. In (-7), replace Latest Updates at -59

60 j with s to obtain H(s)H( s). (-73) n (s/ j ) c As can be seen from inspection of (-73), the poles of H(s)H(-s) are the roots of + (s/j c ) n =. That is, each pole must be one of the numbers c /n p j ( ). (-74) There are n distinct values of p; in what follows, they are denoted as p, p, p,, p n-. These poles are found by multiplying the n roots of - by the complex constant j c. The n roots of - are obtained easily. Complex variable z is a n root of - if n z { k}, k an integer. (-75) Clearly, z must have unity magnitude and phase /n, modulo /n. Hence, the n roots of - form the set Im to j Region of Convergence for H(s) to Re to j Figure -3: Shaded region depicts Re[s] >, region of convergence for transform H(s), the s-domain transfer function of a n th -order Butterworth filter. Value < depends on c and n. All poles of H(s) must have a real part that is less than, or equal to Latest Updates at -6