2.2.6. 4 Fourier transforms ofpower signals Solution: The function has the form x()=∑δ(t-k7) 2.118) Expand it in a Fourier series as ()=∑ Where T x(t)e 6(t)e= the fourier series representation of the unit impulse train is x()= ∑ Naot (2.119)
Solution: The function has the form: Expand it in a Fourier series as where the Fourier series representation of the unit impulse train is 2.2.6.4 Fourier transforms of power signals =− = − n x(t) (t k T ) (2.118) − = jn t n x t C e 0 ( ) = = = − − − − T t e dt T x t e dt T C j n t T T j n t n 1 ( ) 1 ( ) 1 0 2 0 2 =− = n jn t e T x t 0 1 ( ) (2.119)
2.2.6. 4 Fourier transforms ofpower signals Take the fourier transform on both sides of (2.119), X(o)=H∑e 2兀 X()= ∑δ(0-moo) That is ∑(-kD)4O∑6(0-nO) (2.120) X x(a\2兀 -37-2T-T 0 T 27 32 2a, t 20o 30. a Periodic impulse sequence and its frequency spectrum
Take the Fourier transform on both sides of (2.119), That is, 2.2.6.4 Fourier transforms of power signals ] 1 ( ) [ 0 =− = n jn t e T X F =− = − = n T n T X 2 ( ), 2 ( ) 0 0 =− =− − − n n (t k T) ( n ) 0 0 (2.120) Periodic impulse sequence and its frequency spectrum
2.2.7 Random signal description 2.2.7.1 Introduction Ua signal is said to be random if it depends on probabilistic laws. Such signals have unpredictable instantaneous values cannot be desired by analytical time models; can be characterized by their statistical and spectral properties UAn observed random signal must be seen as a particular experimental realization of a set (referred to as the ensemble)of similar signals that can all be produced by the same phenomenon or stochastic process
❑A signal is said to be random if it depends on probabilistic laws. Such signals: – have unpredictable instantaneous values; – cannot be desired by analytical time models; – can be characterized by their statistical and spectral properties. ❑An observed random signal must be seen as a particular experimental realization of a set (referred to as the ensemble) of similar signals that can all be produced by the same phenomenon or stochastic process. 2.2.7 Random signal description 2.2.7.1 Introduction
2.2.7. Introduction sAmple function: the record of every long-time observation for the time process of a random signal, denoted by x(t)(ig 248) Fig. 2. 48 Random process and sample function
❑Sample function: the record of every long-time observation for the time process of a random signal, denoted by x(t)(Fig. 2.48). 2.2.7.1 Introduction Fig. 2.48 Random process and sample function
2.2.7. Introduction asample record: a sample function in a finite time interval URandom process: the set(or ensemble of all sample functions under the same experimental conditions that is x(}={x(t),x2(1)…,x()2…} (2.121)
❑Sample record: a sample function in a finite time interval. ❑Random process: the set (or ensemble) of all sample functions under the same experimental conditions, that is, 2.2.7.1 Introduction x(t)= x1 (t), x2 (t), , xi (t), (2.121)