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. 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) t x(t)x(t) t x(t)x(t) AnalogDigital

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

5. Deterministic and Random Waveforms  A Deterministic waveform can be modeled as a completely specified function of time.  A Random 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. Block Diagram of A Communication System  All communication systems contain three main sub systems: 1.Transmitter 2.Channel 3.Receiver Transmitter 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 (I j ) The information sent from a digital source (I j ) when the jth massage is transmitted is given by: where P j is the probability of transmitting the j th 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. The Math model in this example violates the following rules: 1.Continuity 2.Finite duration Physical Waveform Mathematical Model Waveform

15. Time Average Operator  Definition: The time average operator is given by,  The operator is a linear operator, 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 periodic with period T 0 if, w(t) = w(t + T 0 ) for all t where T 0 is the smallest positive number that satisfies this relationship  A sinusoidal waveform of frequency f 0 = 1/T 0 Hertz is periodic  Theorem: If the waveform involved is periodic, the time average operator can be reduced to where T 0 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 t 1 to t 2, 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 power is:

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 the square root of the normalized power is the rms value. Definition. The average normalized power is given as follows, Where w(t) is the voltage or current waveform

21. Energy and Power Waveforms  Definition: w(t) is a power waveform if and only if the normalized average power P is finite and nonzero (0 < P < ∞).  Definition: The total normalized energy is  Definition: w(t) is an energy waveform if 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) = So, definition is equivalent to And, Noise Power (N) =

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 sinusoidal waveform. That is, where the phasor c = |c|e j  c and 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. Time Domain Description (Inverse FT) Frequency Domain Description (FT)

32. Fourier Transform - Sufficient Conditions  The waveform w(t) is Fourier transformable if it satisfies both Dirichlet conditions: 1)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. 2)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: 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. Finite Energy

33. 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 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. Substitute -f =

36. Properties of Fourier Transforms Corollaries of 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.  Spectral symmetry of real signals. If w(t) is real, then:

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 = f o and f = -f o due to multiplication with the sinusoidal.

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

43. Dirac Delta Function  Definition: The Dirac delta function δ(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 : x (x)(x)

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. 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 fcfc  (f-f c ) Ae j2  f c t  H(f )H(f )  (f-f c ) H(f c )  (f-f c ) H(f c ) e j2  f c t 2Acos(2  f c t)  fcfc  (f-f c ) -fc-fc  (f+f c )

46. Spectrum of a Sine Wave

48. Sampling Function 0 0TsTs 2Ts2Ts 3Ts3Ts -T s -2T s -3T s 1/T s -1/T s  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 T s whereas the period in the frquency domain is 1/ T s.  This function will be used when studying the Sampling Theorem. f t

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

50. Rectangular Pulses

51. Triangular Pulses

52. Spectrum of a Rectangular Pulse Note the inverse relationship between the pulse width T and the zero crossing 1/T  Rectangular pulse is a time window.  FT is a Sa function, infinite frequency content.  Shrinking (сжатие) time axis causes stretching of frequency axis.  Signals cannot be both time-limited and bandwidth-limited.

53. Spectrum of Sa Function  To find the spectrum of a Sa function we can use duality theorem. Duality: W(t)  w(-f) Because Π is an even and real function

54. Spectrum of Rectangular and Sa Pulses

55. Spectrum of a Time Shifted Rectangular Pulse The spectra shown in previous slides are real because the time domain pulse (rectangular pulse) is real and even. If the pulse is offset in time domain to destroy the even symmetry, the spectra will be complex. Let us now apply the Time delay theorem of Table 2.1 to the Rectangular pulse. Time Delay Theorem: w(t-T d )  W(f) e -jωTd We get: T 1

56. Spectrum of a Triangular Pulse  The spectrum of a triangular pulse can be obtained by direct evaluation of the FT integral.  An easier approach is to evaluate the FT using the second derivative of the triangular pulse.  First derivative is composed of two rectangular pulses as shown.  The second derivative consists of the three impulses.  We can find the FT of the second derivative easily and then calculate the FT of the triangular pulse.

57. Spectrum of a Triangular Pulse

58. Table 2.2 Some FT pairs

59. Key FT Properties  Time Scaling; Contracting the time axis leads to an expansion of the frequency axis.  Duality Symmetry between time and frequency domains. “Reverse the pictures”. Eliminates half the transform pairs.  Frequency Shifting (Modulation); (multiplying a time signal by an exponential) leads to a frequency shift.  Multiplication in Time Becomes complicated convolution in frequency. Mod/Demod often involves multiplication. Time windowing becomes frequency convolution with Sa.  Convolution in Time Becomes multiplication in frequency. Defines output of LTI filters: easier to analyze with FTs. h(t)h(t) x(t)x(t) x(t)*h(t)x(t)*h(t) X(f)X(f) X(f)H(f)X(f)H(f) H(f)H(f)

60. Convolution  The convolution of a waveform w 1 (t) with a waveform w 2 (t) to produce a third waveform w 3 (t) which is where w 1 (t) ∗ w 2 (t) is a shorthand notation for this integration operation and ∗ is read “convolved with”. If discontinuous wave shapes are to be convolved, it is usually easier to evaluate the equivalent integral  Evaluation of the convolution integral involves 3 steps. Time reversal of w 2 to obtain w 2 (-λ), Time shifting of w 2 by t seconds to obtain w 2 (-(λ-t)), and Multiplying this result by w 1 to form the integrand w 1 (λ)w 2 (-(λ-t)).

61. Example for Convolution For 0< t < T For t > T

62. Convolution  y(t)=x(t)*z(t)=  x(τ)z(t- τ )d τ Flip one signal and drag it across the other Area under product at drag offset t is y(t). z(2-  ) 2 tt+1t-1 z(t-  ) 01 x(t)x(t) 01 z(t) t t 01   -6 z(-6-  ) 01 y(t)y(t) -2 2 2 t z(-2-  ) z(0-  ) z(4-  ) -6 -4 z()z()  x()x() 

63. Fourier Transform and Spectra Topics:  Spectrum by Convolution  Spectrum of a Switched Sinusoid  Power Spectral Density  Autocorrelation

64. Spectrum of a triangular pulse by convolution  The tails of the triangular pulse decay faster than the rectangular pulse. WHY ??

65. Spectrum of a Switched Sinusoid Spectrum of a Switched Sinusoid Using the Frequency Translation Property of the Fourier Transform We can get a similar result using the convolution property of the Fourier Transform.

66. Spectrum of a Switched Sinusoid

67. Power Spectral Density (PSD)  We define the truncated version (Windowed) of the waveform by: The average normalized power from the time domain: Using Parseval’s theorem to calculate power from the frequency domain

68. Power Spectral Density  Definition: The Power Spectral Density (PSD) for a deterministic power waveform is where w T (t) ↔ W T (f) and P w (f) has units of watts per hertz. The PSD is always a real nonnegative function of frequency. PSD is not sensitive to the phase spectrum of w(t) The normalized average power is This means the area under the PSD function is the normalized average power.

69. Autocorrelation Function  Definition: The autocorrelation of a real (physical) waveform is Wiener-Khintchine Theorem: PSD and the autocorrelation function are Fourier transform pairs; The PSD can be evaluated by either of the following two methods: 1.Direct method: by using the definition, 2.Indirect method: by first evaluating the autocorrelation function and then taking the Fourier transform: P w (f)= ℑ [R w (τ) ] The average power can be obtained by any of the four techniques.

70. PSD of a Sinusoid

71.  The average normalized power may be obtained by using:

72. Orthogonal Representation, Fourier Series and Power Spectra  Orthogonal Series Representation of Signals and Noise Orthogonal Functions Orthogonal Series  Fourier Series. Complex Fourier Series Quadrature Fourier Series Polar Fourier Series Line Spectra for Periodic Waveforms Power Spectral Density for Periodic Waveforms

73. Orthogonal Functions  Definition : Functions ϕ n (t) and ϕ m (t) are said to be Orthogonal with respect to each other the interval a < t < b if they satisfy the condition, where δ nm is called the Kronecker delta function. If the constants K n are all equal to 1 then the ϕ n (t) are said to be orthonormal functions.

74. Example 2.11 Orthogonal Complex Exponential Functions

75. Orthogonal Series  Theorem: Assume w(t) represents a waveform over the interval a < t <b. Then w(t) can be represented over the interval (a, b) by the series where, the coefficients a n are given by following where n is an integer value : If w(t) can be represented without any errors in this way we call the set of functions {φ n } as a “Complete Set” Examples for complete sets: Harmonic Sinusoidal Sets {Sin(nw 0 t)} Complex Expoents {e jnwt } Bessel Functions Legendare polynominals

76. Orthogonal Series Proof of theorem: Assume that the set {φ n } is sufficient to represent the waveform w(t) over the interval a < t <b by the series We operate the integral operator on both sides t o get, Now, since we can find the coefficients a n writing w(t) in series form is possible. Thus theorem is proved.

77. Application of Orthogonal Series  It is also possible to generate w(t) from the ϕ j (t) functions and the coefficients a j.  In this case, w(t) is approximated by using a reasonable number of the ϕ j (t) functions. w(t) is realized by adding weighted versions of orthogonal functions

78. Ex. Square Waves Using Sine Waves. http://www.educatorscorner.com/index.cgi?CONTENT_ID=2487 n =1 n =3 n =5

79. Fourier Series Complex Fourier Series  The frequency f 0 = 1/T 0 is said to be the fundamental frequency and the frequency nf 0 is said to be the nth harmonic frequency, when n>1.

80. Some Properties of Complex Fourier Series

82. Quadrature Fourier Series  The Quadrature Form of the Fourier series representing any physical waveform w(t) over the interval a < t < a+T 0 is, where the orthogonal functions are cos(nω 0 t) and sin(nω 0 t). Using we can find the Fourier coefficients as:

83. Quadrature Fourier Series Since these sinusoidal orthogonal functions are periodic, this series is periodic with the fundamental period T 0. The Complex Fourier Series, and the Quadrature Fourier Series are equivalent representations. This can be shown by expressing the complex number c n as below For all integer values of n and Thus we obtain the identities and

84. Polar Fourier Series The POLAR F Form is where w(t) is real and The above two equations may be inverted, and we obtain

85. Polar Fourier Series Coefficients

86. Line Spetra for Periodic Waveforms Theorem: If a waveform is periodic with period T 0, the spectrum of the waveform w(t) is where f 0 = 1/T 0 and c n are the phasor Fourier coefficients of the waveform Proof: Taking the Fourier transform of both sides, we obtain Here the integral representation for a delta function was used.

87. Line Spectra for Periodic Waveforms Theorem: If w(t) is a periodic function with period T 0 and is represented by Where, then the Fourier coefficients are given by: The Fourier Series Coefficients can also be calculated from the periodic sample values of the Fourier Transform.

88. Line Spectra for Periodic Waveforms h(t)h(t) The Fourier Series Coefficients of the periodic signal can be calculated from the Fourier Transform of the similar nonperiodic signal. The sample values for the Fourier transform gives the Fourier series coefficients.

89. Single Pulse Continous Spectrum Periodic Pulse Train Line Spectrum Line Spectra for Periodic Waveforms

90. Ex. 2.12 Fourier Coeff. for a Periodic Rectangular Wave

91.  T Sa(  fT) Now compare the spectrum for this periodic rectangular wave (solid lines) with the spectrum for the rectangular pulse. Note that the spectrum for the periodic wave contains spectral lines, whereas the spectrum for the nonperiodic pulse is continuous. Note that the envelope of the spectrum for both cases is the same |(sin x)/x| shape, where x=  Tf. Consequently, the Null Bandwidth (for the envelope) is 1/T for both cases, where T is the pulse width. This is a basic property of digital signaling with rectangular pulse shapes. The null bandwidth is the reciprocal of the pulse width.  Now evaluate the coefficients from the Fourier Transform Ex. 2.12 Fourier Coeff. for a Periodic Rectangular Wave

92. Single Pulse Continous Spectrum Periodic Pulse Train Line Spectrum Ex. 2.12 Fourier Coeff. for a Periodic Rectangular Wave

93. Normalized Power Theorem: For a periodic waveform w(t), the normalized power is given by: where the {c n } are the complex Fourier coefficients for the waveform. Proof: For periodic w(t), the Fourier series representation is valid over all time and may be substituted into Eq.(2-12) to evaluate the normalized power:

94. Power Spectral Density for Periodic Waveforms Theorem: For a periodic waveform, the power spectral density (PSD) is given by where T 0 = 1/f 0 is the period of the waveform and {c n } are the corresponding Fourier coefficients for the waveform. PSD is the FT of the Autocorrelation function

95. Power Spectral Density for a Square Wave The PSD for the periodic square wave will be found. Because the waveform is periodic, FS coefficients can be used to evaluate the PSD. Consequently this problem becomes one of evaluating the FS coefficients.

96. END