Signals and Systems: Part 2
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1 Signals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms Some important Fourier transform theorems Convolution and Modulation Ideal filters
2 Fourier transform definitions In EE 02A, the Fourier transform and its inverse were defined by G(jω) = g(t)e jωt dt, g(t) = 2π G(jω)e jωt dω We used jω to make all transforms similar in form (Fourier, Laplace, DTFT, z). In this class we use 2πf instead of ω. If we replace ω with 2πf, G(f) = g(t)e j2πft dt, g(t) = G(f)e j2πft df The transform and its inverse are almost symmetric; only the sign in the complex exponential changes. This will simplify a lot of the transforms and theorems.
3 Fourier transform existence If g(t) is absolutely integrable, i.e., g(t) dt < then G(f) exists for every frequency f and is continuous. If g(t) has finite energy, i.e., g(t) 2 dt < then G(f) exists for most frequencies f and has finite energy. If g(t) is periodic and has a Fourier series, then G(f) = n= G(nf 0 )δ(f nf 0 ) is a weighted sum of impulses in frequency domain.
4 Fourier transform examples One-sided exponential decay is defined by e at u(t) with a > 0: { 0 t < 0 g(t) = e at t > 0 The Fourier transform of one-sided decay is (simple calculus): G(f) = = 0 0 e at e j2πft dt e (a+j2πf)t dt [ e (a+j2πf)t = (a+j2πf) ] t= t=0 Since g(t) has finite area, its transform is continuous. = a+j2πf Even though g(t) is real, its transform is complex valued. This is the usual situation.
5 Fourier transform examples (cont.) We can rationalize G(f): G(f) = a+j2πf = a j2πf a 2 +4π 2 f 2 = a a 2 +4π 2 f 2 j 2πf a 2 +4π 2 f 2 We can better picture G(f) using polar representation. G(f) = a 2 +4π 2 f, θ G(f) = G = tan ( 2πf ) 2 a
6 Fourier transform examples (cont.) Fourier transform at f = 0: G(0) = g(t)e j2π 0 t dt = g(t) dt In general, G(0) is the area under g(t), called the DC value. For g(t) = e at u(t), G(0) = /a is the only real value. It is also largest in magnitude. For one-sided decay, G( f) = G (f), complex conjugate of G(f). This is true for all real-valued signals. ( G ) (f) = g(t)e j2πft dt = g(t) (e j2πft ) dt = g(t)e j2πft dt = g(t)e j2π( f)t dt = G( f) We need look at only positive frequencies for real-valued signals.
7 Duality Fourier inversion theorem: if g(t) has Fourier transform G(f), then G(f) = g(t)e j2πft dt and g(t) = G(f)e j2πft df Inverse transform differs from forward transform only in sign of exponent. Suppose we consider G( ) to be a function of t instead of f. If we apply the Fourier transform again: G(t)e j2πft dt = G(t)e j2π( f)t dt = g( f) We can summarize this as F{g(t)} = G(f) F{G(t)} = g( f) This is the principle of duality.
8 Important Fourier transforms The unit rectangle function Π(t) is defined by { t < 2 Π(t) = 0 t > 2 Its Fourier transform is F {Π(t)} = = By duality, we know that /2 /2 e i2πft Π(t)dt e i2πft dt = sinπf πf F{sinc(πt)} = Π( f) = Π(f) = sinc(πf) since Π(f) is even. We say Π(t) and sinc(πf) are a Fourier pair: Π(t) sinc(πf) Fact: every finite width pulse has a transform with unbounded frequencies.
9 Fourier transform time scaling If a > 0 and g(t) is a signal with Fourier transform G(f), then F {g(at)} = g(at)e j2πft dt = g(u)e j2πf(u/a)du a = a G( f a If a < 0, change of variables requires reversing limits of integration: F {g(at)} = ( f ) a G a Combining both cases: F {g(at)} = ( f ) a G a Special case: a =. The Fourier transform of g( t) is G( f). Compressing in time corresponds to expansion in frequency (and reduction in amplitude) and vice versa. The sharper the pulse the wider the spectrum. )
10 Fourier transform time scaling example The transform of a narrow rectangular pulse of area is { } F τ Π(t/τ) = sinc(πτf) In the limit, the pulse is the unit impulse, and its tranform is the constant. We can find the Fourier transform directly: so F{δ(t)} = δ(t)e j2πft dt = e j2πft t=0 = δ(t) The impulse is the mathematical abstraction of signal whose Fourier transform has magnitude and phase 0 for all frequencies. By duality, F{} = δ(f). All DC, no oscillation. δ(f)
11 Important Fourier transforms (cont.) Shifted impulse δ(t t 0 ): F{δ(t t 0 )} = δ(t t 0 )e j2πft dt = e j2πft 0 This is a complex exponential in frequency. δ(t t 0 ) e j2πft 0 This is an example of shift theorem: F{f(t t 0 )} = e j2πft 0 F{f(t)}. By duality, complex exponential in time has an impulse in frequency: e j2πf0t δ(f f 0 ) Sinuoids: frequency content is concentrated at ±f 0 Hz. F{cos2πf 0 t} = F{ 2 (ej2πf 0t +e j2πf0t )} = 2 δ(f f 0)+ 2 δ(f +f 0) F{sin2πf 0 t} = F{ 2i (ej2πf 0t +e j2πf0t )} = 2i δ(f f 0) 2i δ(f +f 0)
12 Important Fourier transforms (cont.) Laplacian pulse g(t) = e a t where a > 0. Since we can use reversal and additivity: G(f) = g(t) = e at u(t)+e at u( t), a+j2πf + a j2πf = 2a a 2 +4π 2 f 2. This is twice the real part of the Fourier transform of e at u(t) The signum function can be approximated as ( sgn(t) = lim e at u(t) e at u( t) ) a 0 This has the Fourier transform ( F {sgn(t)} = lim a 0 a+j2πf a j2πf ) = jπf
13 Important Fourier transforms (cont.) The unit step function is This has the Fourier transform u(t) = sgn(t) F{u(t)} = 2 δ(f)+ j2πf
14 Fourier transform properties Time delay causes linear phase shift. F {g(t t 0 )} = = = e j2πft 0 g(t t 0 )e j2πft dt g(u)e j2πf(u+t 0) du g(u)e j2πfu du = e j2πft 0 G(f) Therefore g(t t 0 ) e j2πft 0 G(f) By duality we get frequency shifting (modulation): e j2πfct g(t) G(f f c ) cos(2πf c t)g(t) 2 G(f +f c)+ 2 G(f f c)
15 Fourier transform properties (cont.) Convolution in time. The convolution of two signals is g (t) g 2 (t) = g (u)g 2 (t u)du The Fourier transform of the convolution is the product of the transforms. g (t) g 2 (t) G (f)g 2 (f) The Fourier transform reduces convolution to a simpler operation. Note that there is no factor of as there was in EE 02A. 2π for frequency domain convolution,
16 Fourier transform properties (cont.) Multiplication in time. Again, there is no 2π g (t)g 2 (t) G (λ)g 2 (f λ)dλ factor, as there was in EE 02A. Convolution in one domain goes exactly to multiplication in the other domain, and multiplication to convolution. The modulation theorem is a special case. F {g(t)cos(2πf c t)} = F {g(t))} F {cos(2πf c t)} = G(f) ( 2 δ(f +f 0)+ 2 δ(f f 0) ) = 2 G(f +f c) + 2 G(f f c)
17 The triangle function (t) and its Fourier transform The book defines the triangle function (t) as { 2 x x (t) = 2 0 otherwise This is another unfortunate choice, but not as bad as sinc(t)! The triangle function can be written as twice the convolution of two rectangle functions of width 2. (t) = 2 Π(2t) Π(2t) where the factor of 2 is needed to make the convolution at t = 0. The Fourier transform is therefore Then F { (t)} = F {2Π(2t) Π(2t)} = 2F ( {Π(2t)} ) 2 = 2 ( 2 sinc( π 2 f)) 2 = 2 sinc2( 2 πf) (t) 2 sinc2 ( 2 πf)
18 Modulation theorem Modulation of a baseband signal creates replicas at ± the modulation frequency.
19 Applications of modulation For transmission by radio, antenna size is proportional to wavelength. Low frequency signals (voice, music) must be converted to higher frequency. To share bandwidth, signals are modulated by different carrier frequencies. North America AM radio band: KHz (0 KHz bands) North America FM radio band: MHz (200 KHz bands) North America TV bands: VHF 54 72, 76 88, 74 26, UHF , Frequencies can be reused in different geographical areas. With digital TV, channel numbers do not correspond to frequencies. Rats laughing:
20 Bandpass signals Bandlimited signal: G(f) = 0 if f > B. Every sinusoid sin(2πf c t) has bandwidth f c. If g c (t) and g s (t) are bandlimited, then m(t) = g c (t)cos(2πf c t)+g s (t)sin(2πf c t) is a bandpass signal. Its Fourier transform or spectrum is restricted to f c B < f < f c +B The bandwidth is (f c +B) (f c B) = 2B. Most signals of interest in communications will be either bandpass (RF), or baseband (ethernet).
21 Filters A filter is a system that modifies an input. (E.g., an optical filter blocks certain frequencies of light.) In communication theory, we usually consider linear filters, which are linear time-invariant systems. L(v (t)+v 2 (t)) = L(v (t))+l(v 2 (t)) L(av(t)) = al(v(t)) L(v(t t 0 )) = (Lv)(t t 0 ) Fundamental fact: every LTIS is defined by convolution: Lv(t) = h(t) v(t) = h(u)v(t u)du = The signal h(t) is called the impulse response because Lδ(t) = δ(u)h(t u)du = h(t) h(t u)v(u)du
22 Transfer function By the convolution theorem, H(f) is called the transfer function. h(t) v(t) H(f)V(f) Many systems are best understood in the frequency domain. Example: low pass filter with cutoff frequency B: ( f ) H(f) = Π e j2πft d 2Bsinc(2πB(t t d )) 2B Note that the system is not (cannot) be causal.
23 Low-pass filter example x(t) = cos(2π(0.3)t) + cos(2π(0.8)t), h(t) = sinc(πt), y(t) = cos(2π(0.3)t)
24 Butterworth filter: nonideal low-pass filter Butterworth filter has H(f) = (f/b) 2n
25 Butterworth filter vs. ideal low-pass filter
26 High-pass and band-pass filters If the phase is zero h highpass (t) = δ(t) 2Bsinc(2πBt) h bandpass (t) = 2cos(2πf 0 t)2bsinc(2πbt) In practice there will be an additional linear phase in the spectral profile, corresponding to a delay in time. This makes the filters realizable.
27 Band-pass filter example x(t) = cos(2π(0.3)t) + cos(2π(0.8)t), h(t) = cos 2πt sinc πt, y(t) = cos(2π(0.8)t)
28 Low-pass filter: x(t) = Π( t ), h(t) = sinct
29 Low-pass filter: x(t) = Π( t ), h(t) = sinct
30 Band-pass filter: x(t) = Π( t ), h(t) = cos2πt sinct
31 Examples of Communication Channels wires (PCD trace or conductor on IC) optical fiber (attenuation 4dB/km) broadcast TV (50 kw transmit) voice telephone line (under -9 dbm or 0 µw) walkie-talkie: 500 mw, 467 MHz Bluetooth: 20 dbm, 4 dbm, 0 dbm Voyager: X band transmitter, 60 bit/s, 23 W, 34m dish antenna years-later-voyager-offically-exits-the-heliosphere/
32 Communication Channel Distortion The linear description of a channel is its impulse response h(t) or equivalently its transfer function H(f). y(t) = h(t) x(t) Y(f) = H(f)X(f) Note that H(f) both attenuates ( H(f) ) and phase shifts ( H(f)) the input signal. Channels are subject to impairments: Nonlinear distortion (e.g., clipping) Random noise (independent or signal dependent) Interference from other transmitters Self interference (reflections or multipath)
33 Channel Equalization Linear distortion can be compensated for by equalization. H eq (f) = H(f) ˆX(f) = H eq (f)y(f) = X(f) The equalization filter accentuates frequencies that are attenuated by the channel. However, if y(t) includes noise or interference, then Equalization may accentuate noise! y(t) = x(t)+z(t) H eq (f)y(f) = X(f)+ Z(f) H(f)
34 Next time Lab this Friday : How do you figure out where signals are? Spectrograms and waterfall plots Next class Monday : in Lathi and Ding. Signal distortion, power spectral density, correlation and autocorrelation. Wednesday : Begin Chapter 4. Analog modulation schemes. Lab next Friday : finding and decoding airband AM
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