Chapter 2 ●● Signals and spectra
1 Chapter 2 Signals and Spectra
Introduction Basic signal properties(dc, rms, dBm, and power Fourier transform and spectra Linear systems and linear distortion Bandlimited signal and sampling Discrete Fourier transform 。 Bandwidth of signal
2 Introduction • Basic signal properties(dc, rms,dBm, and power) • Fourier transform and spectra • Linear systems and linear distortion • Bandlimited signal and sampling • Discrete Fourier transform • Bandwidth of signal
2.1 Properties of signal and Noise (Properties of Physical Waveform) having significant nonzero values over a composite time interval that is finite Its spectrum having significant values over a composite frequency interval that Is fInite being a continuous function of time; having a finite peak value The waveform having only real values. That is, at any time, it cannot have a complex value a+bi where b is nonzero
3 2.1 Properties of signal and Noise (Properties of Physical Waveform) • having significant nonzero values over a composite time interval that is finite; • Its spectrum having significant values over a composite frequency interval that is finite; • being a continuous function of time; • having a finite peak value; • The waveform having only real values. That is, at any time, it cannot have a complex value a+bj, where b is nonzero
2.1 Properties of signal and Noise Wa to zero before to zero before 3T 5T 6T (a) Physical Waveform (t) Waveform extends Waveform extends 5T6 (b)Math Model Waveform Figure 2-1 Physical and mathematical waveforms
4 2.1 Properties of signal and Noise
2.1 Properties of signal and Noise Time average operator T/2 =lim T/2 Periodic waveform with period To O(1)=(t+0) for all t(2-3) Time average operator for periodic waveform T/2)+ T/2)+a
5 2.1 Properties of signal and Noise • Time average operator (2 -1) 1 lim / 2 / 2 dt T T T T • Periodic waveform with period T0 ( ) ( ) for all t (2 - 3) T0 t t • Time average operator for periodic waveform (2 - 4) 1 ( / 2) ( / 2) 0 dt T T a T a
2.1 Properties of signal and Noise De value -T/2 de= lim O(tt(2-5) T→∞T T/2 · Instantaneous power p(t)=v(t)i(t)(2-6) i(t) 一· Average power ircuit vIt P=p(t)=v(ti(t Theorem. If a load is resistive the average power is ()尔 s=l/msR=MiMs(2-12) R R Wherer is value of the resistive load
6 2.1 Properties of signal and Noise • Dc value ( ) (2 - 5) 1 lim / 2 / 2 t dt T W T T T dc • Instantaneous Power p(t) v(t)i(t) (2 - 6) P p(t) v(t)i(t) (2 - 7) • Average Power ( ) (2 -12) ( ) 2 2 2 2 rms rms rms rms I R V I R V i t R R v t P circuit i(t) v(t) • Theorem. If a load is resistive, the average power is : • Where R is value of the resistive load
2.1 Properties of signal and Noise Rms value Periodic waveform with period To O(t)=o(t+10) Time average operator for periodic waveform T/2)+a /2)+a olt
7 2.1 Properties of signal and Noise • Rms Value ( ) 2 W t rms • Periodic waveform with period T0 ( ) ( ) T0 t t • Time average operator for periodic waveform dt T T a T a ( / 2) ( / 2) 0 1
2.1 Properties of signal and Noise · Example2-1(p37) (a)oltage (e) Instantaneous Powe Figure 2-3 Steady-state waveshapes for Example 2-1
8 2.1 Properties of signal and Noise • Example 2-1(p37)
2.1 Properties of signal and Noise Average normalized power 1r7/2 P=(0(t)=lim 7>7m/2O() total normalized energy T/2 7→7/2O2(t)dl E= lim
9 2.1 Properties of signal and Noise • Average normalized power / 2 / 2 2 2 ( ) 1 ( ) lim T T T t dt T P t • total normalized energy / 2 / 2 2 lim ( ) T T T E t dt
2.1 Properties of signal and Noise Power waveform o(t)is a power waveform if and only if the average normalized power P is finite and nonzero(i.e. 0<P<oo) Energy waveform o(t)is a energy waveform if and only if the average total normalized energy e is finite and nonzero (i. e O<E<oo)
10 2.1 Properties of signal and Noise • Power waveform ω(t) is a power waveform if and only if the average normalized power P is finite and nonzero (i.e. 0<P<∞) • Energy waveform ω(t) is a energy waveform if and only if the average total normalized energy E is finite and nonzero (i.e. 0<E<∞)
