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) the waveform has significant nonzero values over a composite time interval that is finite The spectrum of the waveform has significant values over a composite frequency interval that is finite The waveform is a continuous function of time The waveform has a finite peak value The waveform has only real values. That is, at any time, it cannot have a complex value atbi where b is nonzero
3 2.1 Properties of signal and Noise (Properties of Physical Waveform) • the waveform has significant nonzero values over a composite time interval that is finite • The spectrum of the waveform has significant values over a composite frequency interval that is finite • The waveform is a continuous function of time • The waveform has a finite peak value • The waveform has 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 (t) Waveform decays Waveform decays to zero before to zero before t =+oo 5T 6T (a)Physical Waveform 0(t Waveform extends Waveform extends 5T 6T () )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 少= limD(2 Periodic waveform with period To a(t=a(t+To) for allt (2-3) Time average operator for periodic waveform 1r(T/2)+a T/2)+a 5
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 d t T T a T a + − + • = •
2.1 Properties of signal and Noise De value dc 7-_ T12O(tydr =im (2-5) · Instantaneous power p(t)=v()()(2-6) i(t) e. Average power circuit P=(p( Theorem. If a load is resistive the average power is v(t (OR=-rms R=1 (2-12) R R rs Where r 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 d c → − = • 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 rs Periodic waveform with period To O(t)=o(t+70) e. Time average operator for periodic waveform 少=1 (T/2)+a T/2)+a lt 0 7
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-(p37) ) (an) voltage (b) current 人/Nn ce) Instantaneous Power 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 P=(o(D)=li T/2 o(tdt T→>∞T total normalized eners T/2 E=lim T/2 o(tdt 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. 0<E<oo) 9
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 • 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<∞)
2.1 Properties of signal and Noise Decibel(db this is a base 10 logarithmic measure of a power ratios, for a circuit, O 10 lo average power out s10 out dB/10 dB=10logp average power in /R dB= 10 log rms out out 2 /R rns In -dB=20 log ms-ou +10log Rim or dB-20l0g ms-ou +10 log Rou rs-In Ims-In Rn In Engineering practice dB=20 logl rms-out rms-out 10 rs-ln rms-In
10 2.1 Properties of signal and Noise • Decibel(dB) this is a base 10 logarithmic measure of power ratios, for a circuit, = = power in power out 10 log 10 log average average P P dB i n out = rms i n i n rms out out V R V R dB / / 10 log 2 _ 2 _ + = + = − − − − i n out rms i n rms out out i n rms i n rms out R R I I dB R R V V dB 20 log 10 log or 20 log 10 log /10 10dB in out P P = • In Engineering Practice: = = − − − − rms i n rms out rms i n rms out I I V V dB 20 log 20 log
