Communication Theory

Communication Theory - ECEN 4242 Communication Theory B....

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ECEN 4242 Communication Theory B. Derras University of Colorado at Boulder, ECE Department 1 9/30/01 Communication Theory ECEN 4242 Frequently Asked Questions Chapter 1 & Chapter 2 Q1 When do we use Fourier transform and when do we use Fourier series? A1 Both Fourier transform and Fourier series are used to determine the frequency content of a give signal. However, Fou- rier transform is used when the given signal is aperiodic (non-periodic), whereas Fourier series is used when the given signal is periodic with a given period. The frequency content of a non-periodic signal is continuous. The frequency content of a periodic signal is discrete, i.e. the frequency components of a periodic signal are given only at speciﬁc frequencies that are multiple of the fundamental frequency of the periodic signal. For instance, if the periodic signal has a period T 0 and frequency f 0 =1/ T 0 which is the fundamental frequency, then the frequency content of such a sig- nal is deﬁned only at f = f 0 , 2 f 0 , 3 f 0 , . .., nf 0 , . .... Q2 Is there any physical meaning to negative frequencies? A2 In real-life applications we always deal with positive frequencies. The negative frequencies appear in mathematical formulation of frequency contents of signals and they are used for convenience only. The negative frequencies come from the use of exponential functions to replace trigonometric functions. This is always done for convenience. For instance, let a signal consisting of several sinusoids such as x ( t )= A 1 cos(2 π f 1 t )+ A 2 cos(2 π f 2 t ) + . .. + A n cos(2 π f n t ). It is clear that the frequency content of x ( t ) consists of components located at real frequencies f 1 , f 2 , ..., f n . But since each one of these components can be decomposed into two components with positive and negative frequencies as A i cos(2 π f i t A i /2[exp( j 2 π f i t ) + exp(- j 2 π f i t )], i = 1, 2, . .., n , the resulting frequency content will have components at positive and negative frequencies. One can notice that the amplitude of the real frequency component A i is divided in two halves between the negative and positive components. Therefore from the power or energy standpoint there is no difference between the two representations. Q3 Given a signal x ( t ). To describe its frequency content, when should we use the PSD and when should we use Fourier transform? A3 The power spectral density (PSD) function is used to describe the frequency content of wide-sense stationary or (strictly) stationary random processes, whereas Fourier transform is used to describe the frequency content of deter- ministic signals. This is simply because Fourier transform of a random process is also a random process and therefore it cannot be used to describe something deterministic such as the frequency content. Q4 What is the relationship between the PSD and Fourier transform of a signal x ( t )?

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ECEN 4242 Communication Theory B. Derras University of Colorado at Boulder, ECE Department 2 9/30/01 A4 For a deterministic signal x ( t ), its PSD S x ( f ) and Fourier transform X ( f ) are related by S x ( f )= X ( f ) X ( f )*=| X ( f )| 2 (the symbol * designates complex conjugate). But for a random process (which must be at least wide-sense stationary),
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Communication Theory - ECEN 4242 Communication Theory B....

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