Lecture 1.3. Signals. Fourier Transform.

1. Lecture 1.3.Signals. Fourier Transform.

2. What is a communication system? • Communication systems are designed to transmit information. • Communication systems design concerns: • Selection of the information–bearing waveform; • Bandwidth and power of the waveform; • Effect of system noise on the received information; • Cost of the system.

3. x(t) t x(t) t Analog Digital Digital and Analog Sources and Systems Basic Definitions: • Analog Information Source: An analog information source produces messages which are defined on a continuum. (E.g. :Microphone) • Digital Information Source: A digital information source produces a finite set of possible messages. (E.g. :Typewriter)

4. Digital and Analog Sources and Systems • A digital communication systemtransfers information from a digital source to the intended receiver (also called the sink). • An analog communication systemtransfers information from an analog source to the sink. • Adigital waveformis defined as a function of time that can have a discrete set of amplitude values. • An Analog waveformis a function that has a continuous range of values.

5. Deterministic and Random Waveforms • ADeterministic waveformcan be modeled as a completely specified function of time. • ARandom Waveform(or stochastic waveform) cannot be modeled as a completely specified function of time and must be modeled probabilistically. • We will focus mainly on deterministic waveforms.

6. Receiver Transmitter Block Diagram of A Communication System • All communication systems contain three main sub systems: • Transmitter • Channel • Receiver

7. What makes a Communication System GOOD • We can measure the“GOODNESS”of a communication system in many ways: • How close is the estimate to the original signal m(t) • Better estimate = higher quality transmission • Signal to Noise Ratio (SNR) for analog m(t) • Bit Error Rate (BER) for digital m(t) • How much power is required to transmit s(t)? • Lower power = longer battery life, less interference • How much bandwidth B is required to transmit s(t)? • Less B means more users can share the channel • Exception: Spread Spectrum -- users use same B. • How much information is transmitted? • In analog systems information is related to B of m(t). • In digital systems information is expressed in bits/sec.

8. Measuring Information • Definition:Information Measure (Ij) The information sent from a digital source (Ij) when the jth massage is transmitted is given by: where Pjis the probability of transmitting the jth message. • Messages that are less likely to occur (smaller value for Pj) provide more information (large value of Ij). • The information measure depends on only the likelihood of sending the message and does not depend on possible interpretation of the content. • For units of bits, the base 2 logarithm is used; • if natural logarithm is used, the units are “nats”; • if the base 10 logarithm is used, the units are “hartley”.

9. Measuring Information • Definition: Average Information(H) The average information measure of a digital source is, • where m is the number of possible different source messages. • The average information is also called Entropy. • Definition: Source Rate (R) The source rate is defined as, • where H is the average information • T is the time required to send a message.

10. Channel Capacity & Ideal Comm. Systems • For digital communication systems, the “Optimum System” may defined as the system that minimize the probability of bit error at the system output subject to constraints on the energy and channel bandwidth. • Is it possible to invent a system with no error at the output even when we have noise introduced into the channel? • Yes under certain assumptions !. • According Shannon the probability of error would approach zero, if R< C Where • R - Rate of information (bits/s) • C - Channel capacity (bits/s) • B - Channel bandwidth in Hz and • S/N - the signal-to-noise power ratio Capacity is the maximum amount of information that a particular channel can transmit. It is a theoretical upper limit. The limit can be approached by using Error Correction

11. Channel Capacity & Ideal Comm. Systems ANALOG COMMUNICATION SYSTEMS In analog systems, the OPTIMUM SYSTEM might be defined as the one that achieves the Largest signal-to-noise ratio at the receiver output, subject to design constraints such as channel bandwidth and transmitted power. Question: Is it possible to design a system with infinite signal-to-noise ratio at the output when noise is introduced by the channel? Answer: No! DIMENSIONALITY THEOREM for Digital Signalling: Nyquist showed that if a pulse represents one bit of data, noninterfering pulses can be sent over a channel no faster than 2B pulses/s, where B is the channel bandwidth.

12. Properties of Signals & Noise • In communication systems, the received waveform is usually categorized into two parts: • Signal: • The desired part containing the information. • Noise: • The undesired part • Properties of waveforms include: • DC value, • Root-mean-square (rms) value, • Normalized power, • Magnitude spectrum, • Phase spectrum, • Power spectral density, • Bandwidth • ………………..

13. Physically Realizable Waveforms • Physically realizable waveforms are practical waveforms which can be measured in a laboratory. • These waveforms satisfy the following conditions • 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+jb, where b is nonzero.

14. Physically Realizable Waveforms • Mathematical Models that violate some or all of the conditions listed above are often used. • One main reason is to simplify the mathematical analysis. • If we are careful with the mathematical model, the correct result can be obtained when the answer is properly interpreted. Physical Waveform Mathematical Model Waveform • The Math model in this example violates the following rules: • Continuity • Finite duration

15. Time Average Operator • Definition: The time average operatoris given by, • The operator is a linearoperator, • the average of the sum of two quantities is the same as the sum of their averages:

16. Periodic Waveforms • Definition A waveform w(t)is periodicwith period T0if, w(t) = w(t + T0)for all t where T0 is the smallest positive number that satisfies this relationship • A sinusoidal waveform of frequency f0 = 1/T0 Hertz is periodic • Theorem: If the waveform involved is periodic, the time average operator can be reduced to where T0 is the period of the waveform and a is an arbitrary real constant, which may be taken to be zero.

17. DC Value • Definition: The DC(direct “current”) value of a waveform w(t) is given by its time average, w(t). Thus, • For a physical waveform, we are actually interested in evaluating the DC value only over a finite interval of interest, say, from t1to t2, so that the dc value is

18. Power • Definition. Let v(t) denote the voltage across a set of circuit terminals, and let i(t) denote the current into the terminal, as shown . The instantaneous power(incremental work divided by incremental time) associated with the circuit is given by: p(t) = v(t)i(t) the instantaneous power flows into the circuit when p(t) is positive and flows out of the circuit when p(t) is negative. • The average poweris:

19. RMS Value • Definition: The root-mean-square (rms)value of w(t) is: • Rms value of a sinusoidal: • Theorem: If a load is resistive (i.e., with unity power factor), the average power is: where R is the value of the resistive load.

20. Normalized Power • In the concept of Normalized Power, R is assumed to be 1Ω, although it may be another value in the actual circuit. • Another way of expressing this concept is to say that the power is given on a per-ohm basis. • It can also be realized that thesquare root of the normalized power is the rms value. Definition. The average normalized poweris given as follows, Where w(t) is the voltage or current waveform

21. Energy and Power Waveforms • Definition: w(t)is a power waveformif and only if the normalized average power P is finite and nonzero (0 < P < ∞). • Definition: The total normalized energyis • Definition: w(t)is an energy waveformif and only if the total normalized energy is finite and nonzero (0 < E < ∞).

22. Energy and Power Waveforms • If a waveform is classified as either one of these types, it cannot be of the other type. • If w(t)has finite energy, the power averaged over infinite time is zero. • If the power (averaged over infinite time) is finite, the energy if infinite. • However, mathematical functions can be found that have both infinite energy and infinite power and, consequently, cannot be classified into either of these two categories. (w(t) = e-t). • Physically realizable waveforms are of the energy type. –We can find a finite power for these!!

23. Decibel • A base 10 logarithmic measure of power ratios. • The ratio of the power level at the output of a circuit compared with that at the input is often specified by the decibel gain instead of the actual ratio. • Decibel measure can be defined in 3 ways • Decibel Gain • Decibel signal-to-noise ratio • Mill watt Decibel or dBm • Definition:Decibel Gain of a circuit is:

24. Decibel Gain • If resistive loads are involved, • Definition of dB may be reduced to, or

25. Decibel Signal-to-noise Ratio (SNR) • Definition. The decibel signal-to-noise ratio (S/R, SNR) is: Where, Signal Power (S) = And, Noise Power (N) = So, definition is equivalent to

26. Decibel with Mili watt Reference (dBm) • Definition. The decibel power level with respect to 1 mW is: = 30 + 10 log (Actual Power Level (watts) • Here the “m” in the dBm denotes a milliwatt reference. • When a 1-W reference level is used, the decibel level is denoted dBW; • when a 1-kW reference level is used, the decibel level is denoted dBk. E.g.: If an antenna receives a signal power of 0.3W, what is the received power level in dBm? dBm = 30 + 10xlog(0.3) = 30 + 10x(-0.523)3 = 24.77 dBm

27. Phasors • Definition: A complex number c is said to be a “phasor” (фазовый вектор)if it is used to represent a sinusoidalwaveform. That is, where the phasor c = |c|ejcand Re{.} denotes the real part of the complex quantity {.}. • The phasor can be written as:

28. Fourier Transform and Spectra Topics: • Fourier transform (FT) of a waveform • Properties of Fourier Transforms • Parseval’s Theorem and Energy Spectral Density • Dirac Delta Function and Unit Step Function • Rectangular and Triangular Pulses • Convolution

29. Fourier Transform of a Waveform • Definition: Fourier transform The Fourier Transform(FT) of a waveform w(t)is: • where ℑ[.] denotes the Fourier transform of [.] • f is the frequency parameter with units of Hz (1/s). • W(f) is also called Two-sided Spectrum of w(t), since both positive and negative frequency components are obtained from the definition

30. Evaluation Techniques for FT Integral • One of the following techniques can be used to evaluate a FT integral: • Direct integration. • Tables of Fourier transforms or Laplace transforms. • FT theorems. • Superposition to break the problem into two or more simple problems. • Differentiation or integration of w(t). • Numerical integration of the FT integral on the PC via MATLAB or MathCAD integration functions. • Fast Fourier transform (FFT) on the PC via MATLAB or MathCAD FFT functions.

31. Fourier Transform of a Waveform • Definition: Inverse Fourier transform The Inverse Fourier transform(FT) of a waveform w(t) is: • The functions w(t) and W(f)constitute a Fourier transform pair. Frequency Domain Description (FT) Time Domain Description (Inverse FT)

32. Fourier Transform - Sufficient Conditions • The waveform w(t) is Fourier transformable if it satisfies both Dirichlet conditions: • Over any time interval of finite length, the function w(t) is single valued with a finite number of maxima and minima, and the number of discontinuities (if any) is finite. • w(t) is absolutely integrable. That is, • Above conditions are sufficient, but not necessary. • A weaker sufficient condition for the existence of the Fourier transform is: Finite Energy • where E is the normalized energy. • This is the finite-energy condition that is satisfied by all physically realizable waveforms. • Conclusion:All physical waveforms encountered in engineering practice are Fourier transformable.

33. Spectrum of an Exponential Pulse

34. Spectrum of an Exponential Pulse

35. Properties of Fourier Transforms • Theorem : Spectral symmetry of real signals If w(t) is real, then Superscript asterisk is conjugate operation. • Proof: Take the conjugate Substitute -f = • Since w(t) is real, w*(t) = w(t), and it follows that W(-f) = W*(f). • If w(t) is real and is an even function of t, W(f) is real. • If w(t) is real and is an odd function of t, W(f) is imaginary.

36. Corollaries of Properties of Fourier Transforms • Spectral symmetry of real signals. If w(t) is real, then: • Magnitude spectrum is even about the origin. • |W(-f)| = |W(f)| (A) • Phase spectrum is odd about the origin. • θ(-f) = - θ(f)(B) Since, W(-f) = W*(f) We see that corollaries (A) and (B) are true.

37. Properties of Fourier Transform • f, called frequency and having units of hertz, is just a parameter of the FT that specifies what frequency we are interested in looking for in the waveform w(t). • The FT looks for the frequency f in the w(t)over all time, that is, over -∞ < t < ∞ • W(f )can be complex, even though w(t)is real. • If w(t)is real, then W(-f) = W*(f).

38. Parseval’s Theorem and Energy Spectral Density • Persaval’s theorem gives an alternative method to evaluate energy in frequency domain instead of time domain. • In other words energy is conserved in both domains.

39. Parseval’s Theorem and Energy Spectral Density The total Normalized Energy E is given by the area under the Energy Spectral Density

40. TABIE 2-1: SOME FOURIER TRANSFORM THEOREMS

41. Example 2-3: Spectrum of a Damped Sinusoid • Spectral Peaks of the Magnitude spectrum has moved to f = foand f = -fodue to multiplication with the sinusoidal.

42. Example 2-3: Spectrum of a Damped Sinusoid Variation of W(f) with f

43. d(x) x Dirac Delta Function • Definition: The Dirac deltafunctionδ(x) is defined by where w(x) is any function that is continuous at x = 0. An alternative definition of δ(x) is: The Sifting Property of the δ function is If δ(x) is an even function the integral of the δ function is given by:

44. Unit Step Function • Definition: The Unit Step function u(t) is: Because δ(λ) is zero, except at λ = 0, the Dirac delta function is related to the unit step function by

45. Ad(f-fc) Ad(f-fc) H(fc)d(f-fc) Aej2pfct d(f-fc) H(f) fc fc H(fc) ej2pfct Ad(f+fc) 2Acos(2pfct) -fc Spectrum of Sinusoids • Exponentials become a shifted delta • Sinusoids become two shifted deltas • The Fourier Transform of a periodic signal is a weighted train of deltas

46. Spectrum of a Sine Wave

47. Spectrum of a Sine Wave

48. Sampling Function • The Fourier transform of a delta train in time domain is again a delta train of impulses in the frequency domain. • Note that the period in the time domain is Tswhereas the period in the frquency domain is 1/ Ts . • This function will be used when studying the Sampling Theorem. t -3Ts -2Ts -Ts 0 Ts 2Ts 3Ts 0 -1/Ts 1/Ts f

49. Fourier Transform and Spectra Topics: • Rectangular and Triangular Pulses • Spectrum of Rectangular, Triangular Pulses • Convolution • Spectrum by Convolution

50. Rectangular Pulses