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1、Chapter 1(Review)Chapter 1(Review)Continuous-timeContinuous-timeSignals and SystemsSignals and Systems1.1 Introduction Any problems about signal analysis and processing may be thought of letting signals trough systems.h(t)f(t)y(t)vFrom f(t) and h(t)，find y(t)， Signal processingvFrom y(t) and h(t)，fi
2、nd f(t) ， Signal reconstructionvFrom f(t) and y(t) ，find h(t) ，System design1.1 Introduction vThere are so many different signals and systems that it is impossible to describe them one by onevThe best approach is to represent the signal as a combination of some kind of most simplest signals which wi
3、ll pass through the system and produce a response. Combine the responses of all simplest signals, which is the system response of the original signal.vThis is the basic method to study the signal analyses and processing.1.2 Continue-time Signal vAll signals are thought of as a pattern of variations
4、in time and represented as a time function f(t).vIn the real-world, any signal has a start. Let the start as t=0 that means f(t) = 0 t0Call the signal causal.Typical signals and their representation vUnit Step u(t) (in our textbook (t)u(t)10tu(t- t0)10tt0vu(t) is basic causal signal, multiply which
5、with any non-causal signal to get causal signal.Typical signals and their representation Sinusoidal Asin(t+) f(t) = Asin(t+)= Asin(2ft+)A - Amplitudef - frequency(Hz)= 2f angular frequency (radians/sec) start phase(radians)Typical signals and their representation vsin/cos signals may be represented
6、by complex exponentialvEulers relationTypical signals and their representation vSinusoidal is basic periodic signal which is important both in theory and engineering.vSinusoidal is non-causal signal. All of periodic signals are non-causal because they have no start and no end.f (t) = f (t + mT) m=0,
7、 1, 2, , Typical signals and their representation vExponential f(t) = et is real 0 , growing sinusoidal 01/a|F(j)|e-at u(t)t1.8 Fourier Transform of typical signalsUnit impulse (t)(t)0tt|F(j)|10 Unit impulse has uniform frequency density in whole frequency range, that means it has infinite wide band
8、.1.9 Fourier Transform of typical signalsConstant 11 2()This result could be got directly based on the symmetry of Fourier Transform.Constant 1 represents direct current signal, and its spectrum is non-zero only at = 0, which is a ()1.9 Fourier Transform of typical signalsSin and cos function Based
9、on the transform pair 1 2() and (t) 1, we have some important conclusions:1.9 Fourier Transform of typical signals-00(j)(-j)Fsin 0t-00()()Fcos 0t1.9 Fourier Transform of typical signalsUnit impulse sequence T 2T-T-2TT(t)t002 0- 0-2 000()00 = 2/T1.10 Properties of Fourier TransformLinear Fourier Tran
10、sform is an integral, and it is a linear operation:If f1(t) F1(j) f2(t) F2(j)Then af1(t) + bf2(t) aF1(j) + bF2(j)1.10 Properties of Fourier TransformTime shiftSignals shift in time domain equals phase shift in frequency domain f (t - t0) F(j)e-jt0Based on the definition of Fourier Transform, the abo
11、ve result is easy to be shown.1.10 Properties of Fourier TransformFrequency shift(Modulation theorem)Modulate carrier sin0t with base band signal f(t)Based on definition of Fourier Transform: f(t)ej0t Fj( 0)1.10 Properties of Fourier TransformWith Eulers relation, it is easy to show: f(t) cos0t = 1/
12、2 f(t) (ej0t +e-j0t ) f(t) cos0t 1/2 Fj( + 0)+Fj( 0) f(t) sin0t j/2 Fj( + 0)-Fj( 0)Spectrum of amplitude modulation F(j)P(t) cos0t1/2 Fj( + 0)+ Fj( 0)/20-0-/2/21P(t)1.10 Properties of Fourier TransformEnergy theoremW is energy of signal, |F(j)|2 named signal energy spectrum is signal energy in unit
13、frequency band that has similar shape with |F(j)|, but no phase information.1.10 Properties of Fourier TransformConvolution theorem convolution theorem in time domainf1(t)*f2(t) F1(j) F2(j) convolution theorem in frequency domainf1(t) f2(t) 1/2 F1(j) *F2(j)1.10 Properties of Fourier TransformTransfe
14、r the convolution operation in one domain to the algebra operation in another domain. Almost all of the properties discussed above could be shown based on the convolution theorem:Time shift: f (t - t0) = f (t )* (t - t0) F(j)e-jt0Frequency shift:f(t)ej0t 1/2Fj( )* 2 ( - 0) =Fj( 0)1.11 Fourier analys
15、es of linear systemH(j) is called system function, or transfer functionh(t)H(j)f(t)y(t)F(j)Y(j)y(t) = f(t) * h(t)Y(j) = F(j) H(j)1.11 Fourier analyses of linear systemvSpectrum analyses is an active research area today. If system is too complicated to be represented by analytical expression, it could be done that input sin signals with different frequencies and measure the system output, then give the system transfer function.
