2 Frequency-Domain Analysis

Transcription

1 2 requency-domain Analysis Electrical engineers live in the two worlds, so to speak, of time and frequency. requency-domain analysis is an extremely valuable tool to the communications engineer, more so perhaps than to other systems analysts. Since the communications engineer is concerned primarily with signal bandwidths and signal locations in the frequency domain, rather than with transient analysis, the essentially steady-state approach of the (complex exponential) ourier series and transforms is used rather than the Laplace transform. 2. Math background 2.. Euler s formula: e jx = cos x + j sin x. () cos (A) = Re { e ja} = 2 ( e ja + e ja) (2) sin (A) = Im { e ja} = 2j ( e ja e ja). (3) ( 2.2. We can use cos x = 2 e jx + e jx) and sin x = 2j many trigonometric identities. See Example 2.4. ( e jx e jx) to derive Example 2.3. Use the Euler s formula to show that d dx sin x = cos x. Example 2.4. Use the Euler s formula to show that cos 2 (x) = 2 (cos(2x) + ). 5

2 2.5. Similar technique gives (a) cos( x) = cos(x), (b) cos ( x 2) π = sin(x), (c) sin 2 x = 2 ( cos (2x)) (d) sin(x) cos(x) = 2 sin(2x), and (e) the product-to-sum formula cos(x) cos(y) = (cos(x + y) + cos(x y)). (4) Continuous-Time ourier Transform Definition 2.6. The (direct) ourier transform of a signal g(t) is defined by + G(f) = g(t)e j2πft dt (5) This provides the frequency-domain description of g(t). Conversion back to the time domain is achieved via the inverse (ourier) transform: g (t) = G (f) e j2πft df (6) We may combine (5) and (6) into one compact formula: G (f) e j2πft df = g (t) G (f) = g (t) e j2πft dt. (7) We may simply write G = {g} and g = {G}. Note that G() = g(t)dt and g() = G(f)df. 6

3 2.7. In some references 5, the (direct) ourier transform of a signal g(t) is defined by In which case, we have 2π Ĝ(ω) = + Ĝ (ω) e jωt dω = g (t) Ĝ (ω) = g(t)e jωt dt (8) g (t) e jωt dt (9) In MATLAB, these calculations are carried out via the commands fourier and ifourier. Note that Ĝ() = g(t)dt and g() = 2π Ĝ(ω)dω. The relationship between G(f) in (5) and Ĝ(ω) in (8) is given by G(f) = Ĝ(ω) ω=2πf () Ĝ(ω) = G(f) f= ω 2π () Before we introduce our first but crucial transform pair in Example 2.2 which will involve rectangular function, we want to introduce the indicator function which gives compact representation of the rectangular function. We will see later that the transform of the rectangular function gives a sinc function. Therefore, we will also introduce the sinc function as well. Definition 2.8. An indicator function gives only two values: or. It is usually written in the form [some condition(s) involving t]. Its value at a particular t is one if and only if the condition(s) inside is satisfied for that t. or example, {, a t a, [ t a] =, otherwise. 5 MATLAB uses this definition. 7

4 Alternatively, we can use a set to specify the values of t at which the indicator function gives the value : {, t A, A (t) =, t / A. In particular, the set A can be some intervals: {, a t a, [ a,a] (t) =, otherwise, and [ a,b] (t) = {, a t b,, otherwise. Example 2.9. Carefully sketch the function g(t) = [ t 5] Definition 2.. Rectangular pulse [3, Ex 2.2 p 45]: (t) = [ t.5] = [.5,.5] (t) This is a pulse of unit height and unit width, centered at the origin. Hence, it is also known as the unit gate function rect (x) [4, p 78]. In [3], the values of the pulse (t) at.5 and.5 are not specified. However, in [4], these values are defined to be.5. In MATLAB, the function rectangularpulse(t) can be used to produce 6 the unit gate function above. More generally, we can produce a rectangular pulse whose rising edge is at a and falling edge is at b via rectangularpulse(a,b,t). ( ) t T = [ ] t T 2 = [ T 2, T 2 ] (t) Observe that T is the width of the pulse. 6 Note that rectangularpulse(-.5) and rectangularpulse(.5) give.5 in MATLAB. 8

5 sinc function Definition 2.. The function sinc(x) (sin x)/x (2) is plotted in igure 2. cos sin / / - sinc igure 2: Sinc function This function plays an important role in signal processing. It is also known as the filtering or interpolating function. The full name of the function is sine cardinal 7. Using L Hôpital s rule, we find lim x sinc(x) =. sinc(x) is the product of an oscillating signal sin(x) (of period 2π) and a monotonically decreasing function /x. Therefore, sinc(x) exhibits sinusoidal oscillations of period 2π, with amplitude decreasing continuously as /x. Its zero crossings are at all non-zero integer multiples of π. 7 which corresponds to the Latin name sinus cardinalis. It was introduced by Woodward in his 952 paper Information theory and inverse probability in telecommunication [], in which he noted that it occurs so often in ourier analysis and its applications that it does seem to merit some notation of its own 9

6 In MATLAB, in [3, p 37], in [2, eq. 2.64], and in [], sinc(x) is defined as (sin(πx))/πx. In which case, its zero crossings are at non-zero integer values of its argument. This is sometimes called the normalized sinc function to distinguish it from (2) which is unnormalized. Example 2.2. Rectangular function and Sinc function: [ t a] sin(2πf a) πf = 2 sin (aω) ω = 2a sinc (aω) (3) sinc 2 ( π ft) f 2 f t ω 2 π f 2 f 2 f 2π f f 2 f ω f T 2 T T 2 t sinc T T ω 2 T 2π T Tsinc ( π T f ) T T ω f igure 3: ourier transform of sinc and rectangular functions By setting a = T /2, we have [ t T 2 ] In particular, when T =, we have rect (t) T sinc(πt f). (4) sinc(πt). transform of the unit gate function is the normalized sinc function. The ourier

7 Definition 2.3. The (Dirac) delta function or (unit) impulse function is denoted by δ(t). It is usually depicted as a vertical arrow at the origin. Note that δ(t) is not 8 a true function; it is undefined at t =. We define δ(t) as a generalized function which satisfies the sampling property (or sifting property) for any function φ(t) which is continuous at t =. φ(t)δ(t)dt = φ() (5) In this way, the delta function has no mathematical or physical meaning unless it appears under the operation of integration. Intuitively we may visualize δ(t) as an infinitely tall, infinitely narrow rectangular pulse of unit area: lim ε ε [ t 2] ε Properties of δ(t): δ(t) = for t. δ(t T ) = for t T. A δ(t)dt = A(). (a) δ(t)dt =. (b) {} δ(t)dt =. (c) x δ(t)dt = [, )(x). Hence, we may think of δ(t) as the derivative of the unit step function U(t) = [, ) (x) [, Defn 3.3 p 26]. g(t)δ(t c)dt = g(c) for g continuous at T. In fact, for any ε >, Convolution 9 property: T +ε T ε (δ g)(t) = (g δ)(t) = gt)δ(t c)dt = g(c). where we assume that φ is continuous at t. g(τ)δ(t τ)dτ = φ(t) (6) 8 The δ-function is a distribution, not a function. In spite of that, it s always called δ-function. 9 See Definition 2.34.

8 δ(at) = a δ(t). In particular, δ(ω) = δ(f) (7) 2π and δ(ω ω ) = δ(2πf 2πf ) = 2π δ(f f ), (8) where ω = 2πf and ω = 2πf. Example δ (t)dt = and 2 δ (t)dt =. Example δ (t)dt = Example 2.7. δ(t). Example 2.8. e j2πf t δ (f f ). Example 2.9. e jω t Example 2.2. e j4πt 2πδ (ω ω ). 2

9 Example 2.2. cos(2πt) cos(2πf t) 2 (δ (f f ) + δ (f + f )). Example cos(2t) Example cos (2πt) + cos (4πt) ourier transform: Example Example cos (t) 3 cos (3t) + 5 cos (5t) t cos 3 cos 3 cos 5 f t Example cos 2 (2πt) 3

10 Example cos (2πt) cos (4πt) Conjugate symmetry : If g(t) is real-valued, then G( f) = (G(f)) (a) Even amplitude symmetry: G ( f) = G (f) (b) Odd phase symmetry: G ( f) = G (f) Observe that if we know X(f) for all f positive, we also know X(f) for all f negative. Interpretation: Only half of the spectrum contains all of the information. Positive-frequency part of the spectrum contains all the necessary information. The negative-frequency half of the spectrum can be determined by simply complex conjugating the positive-frequency half of the spectrum. urthermore, (a) If g(t) is real and even, then so is G(f). (b) If g(t) is real and odd, then G(f) is pure imaginary and odd. Hermitian symmetry in [3, p 48] and [7, p 7 ]. 4

11 2.29. Shifting properties Time-shift: g (t t ) e j2πft G (f) Note that e j2πft =. So, the spectrum of g (t t ) looks exactly the same as the spectrum of g(t) (unless you also look at their phases). requency-shift (or modulation): e j2πft g (t) G (f f ) 2.3. Let g(t), g (t), and g 2 (t) denote signals with G(f), G (f), and G 2 (f) denoting their respective ourier transforms. (a) Superposition theorem (linearity): a g (t) + a 2 g 2 (t) a G (f) + a 2 G 2 (f). (b) Scale-change theorem (scaling property [4, p 88]): ( ) f g(at) a G. a The function g(at) represents the function g(t) compressed in time by a factor a (when a > l). Similarly, the function G(f/a) represents the function G(f) expanded in frequency by the same factor a. 5

12 The scaling property says that if we squeeze a function in t, its ourier transform stretches out in f. It is not possible to arbitrarily concentrate both a function and its ourier transform. Generally speaking, the more concentrated g(t) is, the more spread out its ourier transform G(f) must be. This trade-off can be formalized in the form of an uncertainty principle. See also 2.4 and 2.4. Intuitively, we understand that compression in time by a factor a means that the signal is varying more rapidly by the same factor. To synthesize such a signal, the frequencies of its sinusoidal components must be increased by the factor a, implying that its frequency spectrum is expanded by the factor a. Similarly, a signal expanded in time varies more slowly; hence, the frequencies of its components are lowered, implying that its frequency spectrum is compressed. (c) Duality theorem (Symmetry Property [4, p 86]): G(t) g( f). In words, for any result or relationship between g(t) and G(f), there exists a dual result or relationship, obtained by interchanging the roles of g(t) and G(f) in the original result (along with some minor modifications arising because of a sign change). In particular, if the ourier transform of g(t) is G(f), then the ourier transform of G(f) with f replaced by t is the original timedomain signal with t replaced by f. If we use the ω-definition (8), we get a similar relationship with an extra factor of 2π: Ĝ(t) 2πg( ω). Example 2.3. g(t) = cos(2πaf t) 2 (δ(f af ) + δ(f + af )). 6

13 Example rom Example 2.2, we know that By the duality theorem, we have which is the same as [ t a] 2a sinc (2πaf) (9) 2a sinc(2πat) [ f a], sinc(2πf t) Both transform pairs are illustrated in igure 3. 2f [ f f ]. (2) Example Let s try to derive the time-shift property from the frequencyshift property. We start with an arbitrary function g(t). Next we will define another function x(t) by setting X(f) to be g(f). Note that f here is just a dummy variable; we can also write X(t) = g(t). Applying the duality theorem to the transform pair x(t) X(f), we get another transform pair X(t) x( f). The LHS is g(t); therefore, the RHS must be G(f). This implies G(f) = x( f). Next, recall the frequency-shift property: The duality theorem then gives e j2πct x (t) X (f c). X (t c) ej2πc f x ( f). Replacing X(t) by g(t) and x( f) by G(f), we finally get the time-shift property. Definition The convolution of two signals, g (t) and g 2 (t), is a new function of time, g(t). We write g = g g 2. It is defined as the integral of the product of the two functions after one is reversed and shifted: g(t) = (g g 2 )(t) (2) = + g (µ)g 2 (t µ)dµ = 7 + g (t µ)g 2 (µ)dµ. (22)

14 Note that t is a parameter as far as the integration is concerned. The integrand is formed from g and g 2 by three operations: (a) time reversal to obtain g 2 ( µ), (b) time shifting to obtain g 2 ( (µ t)) = g 2 (t µ), and (c) multiplication of g (µ) and g 2 (t µ) to form the integrand. In some references, (2) is expressed as g(t) = g (t) g 2 (t). Example We can get a triangle from convolution of two rectangular waves. In particular, [ t a] [ t a] = (2a t ) [ t 2a] Convolution theorem: (a) Convolution-in-time rule: g g 2 G G 2. (23) (b) Convolution-in-frequency rule: g g 2 G G 2. (24) Example We can use the convolution theorem to prove the frequencysift property in

15 2.38. rom the convolution theorem, we have g 2 G G if g is band-limited to B, then g 2 is band-limited to 2B Parseval s theorem (Rayleigh s energy theorem, Plancherel formula) for ourier transform: + g(t) 2 dt = + G(f) 2 df. (25) The LHS of (25) is called the (total) energy of g(t). On the RHS, G(f) 2 is called the energy spectral density of g(t). By integrating the energy spectral density over all frequency, we obtain the signal s total energy. The energy contained in the frequency band B can be found from the integral B G(f) 2 df. More generally, ourier transform preserves the inner product [2, Theorem 2.2]: g, g 2 = g (t)g 2(t)dt = G (f)g 2(f)df = G, G (Heisenberg) Uncertainty Principle [2, 9]: Suppose g is a function which satisfies the normalizing condition g 2 2 = g(t) 2 dt = which automatically implies that G 2 2 = G(f) 2 df =. Then ( ) ( t 2 g(t) 2 dt ) f 2 G(f) 2 df 6π2, (26) and equality holds if and only if g(t) = Ae Bt2 where B > and A 2 = 2B/π. In fact, we have ( ) ( t 2 g(t t ) 2 dt for every t, f. ) f 2 G(f f ) 2 df 6π 2, The proof relies on Cauchy-Schwarz inequality. 9

16 t or any function h, define its dispersion h as 2 h(t) 2 dt h(t) 2. Then, we can dt apply (26) to the function g(t) = h(t)/ h 2 and get h H 6π A signal cannot be simultaneously time-limited and band-limited. Proof. Suppose g(t) is simultaneously () time-limited to T and (2) bandlimited to B. Pick any positive number T s and positive integer K such that f s = T s > 2B and K > T T s. The sampled signal g Ts (t) is given by g Ts (t) = k g[k]δ (t kt s ) = K k= K g[k]δ (t kt s ) where g[k] = g (kt s ). Now, because we sample the signal faster than the Nyquist rate, we can reconstruct the signal g by producing g Ts h r where the LP h r is given by H r (ω) = T s [ω < 2πf c ] with the restriction that B < f c < T s B. In frequency domain, we have G(ω) = K k= K g[k]e jkωt s H r (ω). Consider ω inside the interval I = (2πB, 2πf c ). Then, ω>2πb = G(ω) ω<2πf c = T s K k= K g (kt s ) e jkωt s K z=ejωts = T s k= K g (kt s ) z k (27) Because z, we can divide (27) by z K and then the last term becomes a polynomial of the form a 2K z 2K + a 2K z 2K + + a z + a. By fundamental theorem of algebra, this polynomial has only finitely many roots that is there are only finitely many values of z = e jωt s which satisfies (27). Because there are uncountably many values of ω in the interval I and hence uncountably many values of z = e jωt s which satisfy (27), we have a contradiction. 2

17 2.42. The observation in 2.4 raises concerns about the signal and filter models used in the study of communication systems. Since a signal cannot be both bandlimited and timelimited, we should either abandon bandlimited signals (and ideal filters) or else accept signal models that exist for all time. On the one hand, we recognize that any real signal is timelimited, having starting and ending times. On the other hand, the concepts of bandlimited spectra and ideal filters are too useful and appealing to be dismissed entirely. The resolution of our dilemma is really not so difficult, requiring but a small compromise. Although a strictly timelimited signal is not strictly bandlimited, its spectrum may be negligibly small above some upper frequency limit B. Likewise, a strictly bandlimited signal may be negligibly small outside a certain time interval t t t 2. Therefore, we will often assume that signals are essentially both bandlimited and timelimited for most practical purposes. 2