1 Let g(t) be periodic; period = T o. Fundamental frequency = f o = 1/ T o Hz or  o = 2  / T o rad/sec. Harmonics =n f o, n =2,3 4,... Trigonometric forms Communication Systems : Prof. Ravi Warrier EXAMPLE : g(t) t a) For R = 1 M  and C=1 µF, what is g o (t) ? b) For R = 1 M  and C=0.1 µF, what is g o (t) ? 2 FOURIER SERIES

2 Communication Systems : Prof. Ravi Warrier EXERCISE : g(t) t (sec) What is g o (t) if the frequency response of the filter is as shown ? Filter H(  ) g(t)g o (t) EXERCISE : a)Find the Fourier series in trigonometric compact form. g(t) t (sec) 1-2 2

3 Communication Systems : Prof. Ravi Warrier EXAMPLE : g(t) 0 t 2 Sketch the Fourier spectra. Suppose that g(t) is passed through a filter of frequency response as shown. What is the output signal ? (Both positive and negative frequencies are shown here) H(j  ) g(t)g o (t) |H(j  )|   H(j  )

4 Communication Systems : Prof. Ravi Warrier ENERGY AND POWER Energy of g(t): g(t) is an energy signal if E g < . Power of g(t) : g(t) is a power signal if 0< P < . EXAMPLES : Let g(t) be as shown. The energy of g(t) is : E g =54 J. Let g(t) be a unit step function: g(t) = u(t). Is this a power or an energy signal ? g(t) t  1 t  Not an energy signal. A power signal. Average Power of sinewaves POWER OF ANY PERIODIC FUNCTION IN TERMS OF FOURIER COEFFICIENTS : EXAMPLE : g(t) t 

5 Communication Systems : Prof. Ravi Warrier Signal Comparison : CORRELATION Let g 1 (t) and g 2 (t) be two signals. Their correlation is defined as If g 1 (t) = g 2 (t) = g(t), this becomes autocorrelation function, given by We see that  g (0)=E g we ge the signal energy. That is, the signal energy =autocorrelation at  = 0. FOURIER TRANSFORMS Definition : F{ g(t)} = G(  ) F -1 { G(  )} =g(t) TRANSFORM EXAMPLES : g(t) A 0 A 0 AA A 0

6 Communication Systems : Prof. Ravi Warrier g(t)

7 PROPERTIES F { g(t)} = G(  ) 1. Symmetry : 2. Scaling : 3. Time-shifting : 4. Frequency shifting : FOR PROOF READ TEXT. (The term represents a linear phase ; in time domain it is delay). What is Here g(t) is modulating the sinusoid amplitude - AMPLITUDE MODULATION. g(t) is the modulating signal, cos(  o t) is called the carrier. EXAMPLE : We will find the Fourier transform of g(t)cos(10t). 1 0 cos(10t) 0 g(t)cos(10t)  = TIME domain Note : Multiplication in time- domain doesn’t transform to multiplication in frequency domain. 0 Math: EXERCISE : 1. What is 2. Find the spectrum of a) b) Sketch the time functions and the spectra. Communication Systems : Prof. Ravi Warrier FREQUENCY domain

8 Communication Systems : Prof. Ravi Warrier 5. Differentiation EXAMPLE : We find the Fourier transform of the triangular function shown using this property. g(t) A Integration : 7. Convolution : CONVOLUTION IN TIME DOMAIN (  ) MULTIPLICATION IN FREQUENCY DOMAIN CONVOLUTION IN FREQUENCY DOMAIN MULTIPLICATION IN TIME DOMAIN

9 Communication Systems : Prof. Ravi Warrier EXAMPLE: 0 g(t) 0 1 Note : g(t) has a pulse width of  /2 sec but has pulse width of  sec. Convolution increases the width of the function. EXERCISE : Let g(t) = sinc(50t). What is the spectral width (bandwidth) of g(t) ? What is the bandwidth of g 2 (t) ? EXAMPLE : What is the Fourier transform of a periodic function ? Periodic functions can be expressed in time domain as sum of a dc term and sinusoids of fundamental frequency and harmonics. Fourier transform of a sinusoid is a pair of impulse functions. Therefore, the Fourier transform of a periodic function is a sum of impulse functions centered at zero frequency, fundamental frequency and harmonics. EXAMPLE : We find the Fourier transform of the periodic function shown. 1 0 g(t) The Fourier series of g(t) is given (in page 51, text) by The Fourier transform of g(t) is |G(  )|  G(  ) We have

10 Communication Systems : Prof. Ravi Warrier -4T o -3T o -2T o -T o 0 T o 2T o 3T o 4T o g(t) EXAMPLE : Consider the periodic function g(t) consisting of impulse functions at equal spaces of T o sec. We find the Fourier transform of g(t). We can express g(t) as -4  o -3  o -2  o -  o 0  o 2  o 3  o 4  o G(  ) oo SIGNAL ENERGY AND ENEGY SPECTRAL DENSITY We define signal energy as. If g(t) is complex we can express energy as Parseval’s theorem : Signal energy is Energy Spectral Density (ESD): is called the energy spectral density of g(t). The signal energy is the integral of the energy spectral density ( multiplied by 2  ). ESD provides a way of computing energy from the Fourier transform of g(t). EXAMPLE : Consider g(t)=e -0.5t u(t). Find the energy and the ESD of g(t). 0

11 Communication Systems : Prof. Ravi Warrier ENERGY OF MODULATED SIGNAL Let g(t) be a baseband energy signal bandlimited to B Hz. Let Suppose that G(  ) is as shown. ThenG(  ) 0 ()() The signal energy is reduced by 1/2 when it is multiplied by a sinewave of unit amplitude. ESD OF A SYSTEM INPUT AND OUTPUT H(  ) G(  ) Y(  ) EXAMPLE : Find the input and output energies. R=200  and C=0.01 F. g(t)=sinc(t). EXERCISE : Redo the Example problem for R=200  and C=0.001 F.

12 Communication Systems : Prof. Ravi Warrier Autocorrelation Function and ESD : For g(t) a real function Energy Spectral Density is the Fourier Transform of the autocorrelation function. SIGNAL POWER AND POWER SPECTRAL DENSITY(PSD) Energy and energy spectral density are useful for energy signals. For power signals we define power and power spectral density as follows :

13 Communication Systems : Prof. Ravi Warrier Input signal power, output signal power Let g(t) be a power signal applied to a system. H(  ) Y(  ) G(  ) EXAMPLE : Let g(t)=A cos(  o t), a power signal. EXAMPLE : Consider a noise signal n(t) with PSD is the input to a differentiator. What is the output noise power ? H(  )=j  y(t)n(t) K 0

14 Communication Systems : Prof. Ravi Warrier EXERCISE : 1 Consider a noise signal n(t) with PSD applied to a RC filter with RC=1 sec. Determine the input noise power and output noise power. (Input power = 2W, Output power=0.25 W) 2. Suppose that the input to RC filter is g(t)=2cos(0.5t), what is the input and out put signal powers ? (Ans: power in=2W, power out=1.79W) 3. Next consider the input filter to be g(t)+n(t). Find the (signal power)/(noise power) at the input and output. This is called the signal-to-noise ratio. REVIEW : 1) Fourier transform inverse Fourier transform definitions 2) Properties : Important ones :- Symmetry, time-delay /phase shift, Modulation 3) Results : Fourier transform of periodic functions, Energy, ESD, autocorrelation, power, PSD, autocorrelation, input energy-output energy, input power-output power. What is the autocorrelation function of a sinewave ? What is the PSD of a sinewave ? What is the average power of a sinewave ? Does phase shift affect the power and autocorrelation ? What is the autocorrelation function of g(t) =cos(20t) ? What is the autocorrelation function of g(t) =sin(20t) ? READ TEXT BOOK A LOT. Distortionless Transmission : The ideal goal of a communication system is to make sure the received and transmitted signals are the same. That is, the received signal is not distorted. This means the communication system transfer function should have a constant magnitude and linear phase characteristics, in the frequency region of interest. H(  ) Y(  ) G(  ) K 0 0 EXERCISE : Find the Fourier transform and sketch the spectra of : i)g(t) = sinc(20t)cos(100t) ii) g(t) = sinc(20t)cos 2 (100t).

15 g(t)=1 for binary 1, g(t) = -1 for binary 0 Communication Systems : Prof. Ravi Warrier

16 Communication Systems : Prof. Ravi Warrier