Ch. 4. Spectral Density & Correlation 4.1 Energy Spectral Density 4.2 Power Spectral Density 4.3 Time-Averaged Noise Representation 4.4 Correlation Functions 4.5 Properties of Correlation Functions 4.6 Correlation Functions for Finite-Energy Signals 4.7 Band-Limited White Noise
Ch. 4. Spectral Density & Correlation 4.1 Energy Spectral Density Energy Spectral Density For any specific frequency, if is continuous, is zero (density). 즉, 가어떤의미를갖기위해서는반드시 area 가관련되어야한다. f(t) h(t) g(t) [ 그림 4-1] Representation of system 이때, phase term 은계산에서고려하지않는다. 즉 magnitude 만을고려한다. 4-1
[ 그림 4-2] An ideal bandpass filter < 를이런 filter 에넣었을때의 output 을생각해보자. > : real function 즉, 모든 real function 의 energy spectral density 는 even function 이다. : if is real 4-2
< 가주어졌을때실제로 energy spectral density 를찾는방법 > [ 그림 4-3] Measurement of energy spectral density The area under the energy spectral density gives the energy within a given band of frequencies. 4-3
4.2 Power Spectral Density mean squared value of 가주기함수이면 가주기함수가아니면 가필요없지만 가필요하다. power spectral density function. ( 단위 : W/Hz) [ 그림 4-4] (a) A power signal (b)its finite-interval truncation 4-4
Assume : a power signal consider in 이때, 양쪽식의 = 은 결과뿐아니라각 에대해서도만족한다고가정하면 The cumulative power spectrum 이때, phase term 은전혀고려되고있지않다. 즉, 같은 power spectral density 를갖는틀린 가존재할수있다. < conclusion > Power Spectral Density or Power Spectrum or Power Density Spectrum 4-5
< For a periodic signal >, by Parseval's Theorem line power spectrum. A cummulative power spectrum. power spectral density of a periodic function [ 그림 4-5] Power spectra of periodic functions: (a) line power of a periodic function (b) integrated power spectrum of a periodic function (c) power spectral density of a periodic function (d) integrated power spectrum of an aperiodic function (e) power spectral density of an aperiodic function 일반적으로 periodic 성분이있으면 power spectral density 가 impulse 로 나타나며, periodic 성분이없으면 continuous 한형태가된다. 이제는 line power spectrum 을알면 power spectral density 를구할수있다. 4-6
Example 4-1) Find power spectral density of Sol) exponential Fourier Series phase term 은고려하지않음. 4-7
< Note > f(t) h(t) g(t) Power signal Linear system ( time invarient ) [ 그림 4-6] LTI system 에서의 power signal : truncated signals Energy spectral density : Power spectral density : < Conclusion > 두 density 의성격이매우비슷하게나타남을알수있다. 4-8
4.3 Time-Averaged Noise Representation : a noise signal 1. mean value, : dc (or average) value of 2. Mean square value, rms (root mean square) noise power spectral density 3. AC component : ( ) Signal to Noise ratio 4-9
4.4 Correlation Functions Fourier Series를이용하지않고, 곧장 time function에서부터 를구하는방법은? Autocorrelation function of is the inverse of. < conclusion > Transform Pair of Power Spectral Density 4-10
Example 4-2) [ 그림 4-7] (a) A periodic waveform and (b) its autocorrelation function of a periodic fn is also a periodic fn. of aperiodic fn is also aperiodic fn. 4-11
< Autocorrelation function 의용도 > Additive noise 에대한 immunity 즉, additive noise가섞였을때 signal 자체에서는 noise를제거하기가힘들지만 autocorrelation 형태로바꾸게되면쉽게찾아낼수있게된다. [ 그림 4-8] Autocorrelation of a periodic signal plus noise 4-12
< The cross-correlation function > Example 4-3) we want to measure the time delay. [ 그림 4-9] Cross-correlation of a random signal plus noise 4-13
4.5 Properties of Correlation Functions 4.5.1 Symmetry Real part of is even function if is real is real & even. 4.5.2 Mean-Square Value 4.5.3 Periodicity If, 4-14
4.5.4 Average Value : average value of : zero average value : average value of : zero average value 즉, 의 average value 는각, 의 average value 의곱과같다. 4-15
4.5.5 Maximum value 4.5.6 Additivity if (uncorrelated) if x(t) & y(t) are orthogonal (zero average인경우 ) uncorrelated in random variable independent uncorrelation 4-16
4.6 Correlation Functions for Finite-Energy Signals : The autocorrelation function for a signal of finite energy. Autocorrelation Energy spectral density function 4-17
4.7 Band-Limited White Noise for all 1/2 의의미는 two-sided power spectral density 의의미이다. Physically it cannot be present. 그러나, 일반적으로우리가사용하는신호의주파수영역이 보다좁은경우우리는 를 white noise 라가정할수있다. Band Limited White Noise is a zero mean white noise with then n i (t) h(t) n o (t) [ 그림 4-10] Power spectral density of noise signal if is white, 4-18
4.7.1 Thermal Noise T : temperature k : Boltzman 상수 = 1.38 h : Planck's 상수 = 6.625 joule/k joule sec 실제 S f ( f ) kt 2 f [ 그림 4-11] Power spectral density 4-19
< Representation of Bandpass Signals and Systems > narrow bandpass signals and channels : carrier signal보다 bandwidth의폭이많이작은 bandpass signal representation of b-p signals & systems in terms of equivalent lowpass waveforms the characterization of bandpass stationary processes * Representation of Bandpass Signals S ( f ) f c 0 f c f [ 그림 4-12] Spectrum of a bandpass signal : ( 그림과같은 ) a real-valued signal : contains only the positive frequencies in : pre-envelope of or analytic signal 여기서 4-20
여기서 ˆ 이 filter 는입력신호에대해 위상을변환시켜준다. Analytic signal 는 bandpass signal이다. obtain an equivalent lowpass representation. ˆ ˆ 4-21
: usually (in general) a complex signal. baseband signal 로나타낼수있다. 1 desired bandpass representation. ˆ 와 를 의 Quadrature component 라고부른다. 2 where, complex envelope of the real signal. the equivalent lowpass signal. 3 s ( t ) = a ( t ) cos[ 2 π f c t + θ ( t )] envelope phase 4-22
< Fourier Transform of > < Energy in the signal > 이 항은 으로 간다 는 에비해서비교적느리게변화하는신호이다. 그러므로다음과같은그림으로표현되고, 이신호를적분하면거의 0에가깝게되기때문에적분식에서 2번째적분은무시해도좋다. 즉, 신호 의 energy는 의에너지와같다. a 2 ( t ) cos(4π f c t+2θ (t) ) [ 그림 4-13] BandPass 신호의 Envelop 과실제 modulated 신호 4-23