1. Lecture 1.3. Signals. Fourier Transform.

2. 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)
Analog
Digital

3. 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.

4. 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.

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

6. Measuring Information
Definition: Information Measure (Ij)
The information sent from a digital source (Ij) when the jth massage is transmitted is given by:
where Pj is 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”.

7. 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.

8. Channel Capacity & Ideal Comm. Systems
For digital communication systems, the “Optimum System” may be 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

9. 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.
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.

10. 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
………………..

11. 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.

12. 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

13. 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:

14. Periodic Waveforms Definition
A waveform w(t) is periodic with period T0 if,
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.

15. 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 t1 to t2, so that the dc value is

16. 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:

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

18. 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

19. 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)

20. 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.

21. 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)
Corollaries of
Since, W(-f) = W*(f)
We see that corollaries (A) and (B) are true.

22. 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).

23. 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.

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

25. TABIE 2-1: SOME FOURIER TRANSFORM THEOREMS

26. Dirac Delta Function
Definition: The Dirac delta function δ(x) is defined by x
d(x)
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:

27. 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

28. 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 fc
Ad(f-fc)
Aej2pfct 
H(f )
d(f-fc)
H(fc)d(f-fc)
H(fc) ej2pfct
2Acos(2pfct)  fc
Ad(f-fc)
-fc
Ad(f+fc)

29. Spectrum of a Sine Wave

30. Spectrum of a Sine Wave

31. 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 Ts whereas the period in the frquency domain is 1/ Ts .
This function will be used when studying the Sampling Theorem.
-3Ts
-2Ts
-Ts Ts
2Ts
3Ts t
-1/Ts
1/Ts f

32. Rectangular Pulses

33. Spectrum of a Rectangular Pulse
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.
Note the inverse relationship between the pulse width T and the zero crossing 1/T

34. 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

35. Spectrum of Rectangular and Sa Pulses

36. Table 2.2 Some FT pairs

37. 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.
x(t)
x(t)*h(t)
h(t)
X(f)
X(f)H(f)
H(f)

38. Convolution
The convolution of a waveform w1(t) with a waveform w2(t) to produce a third waveform w3(t) which is
where w1(t)∗ w2(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 w2 to obtain w2(-λ),
Time shifting of w2 by t seconds to obtain w2(-(λ-t)), and
Multiplying this result by w1 to form the integrand w1(λ)w2(-(λ-t)).

39. y(t)=x(t)*z(t)=  x(τ)z(t- τ )d τ
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).
x(t)
z(t)
z(t-t)
x(t)
z(t) t t t -1 1 t -1 1 t
t-1 t
t+1
z(-2-t)
z(-6-t)
z(0-t)
z(2-t)
z(4-t) -6 -4 -2 -1 1 2 t 2
y(t) -6 -4 -2 -1 1 2 t

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

41. Power Spectral Density
Definition: The Power Spectral Density (PSD) for a deterministic power waveform is
where wT(t) ↔ WT(f) and Pw(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.

42. 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:
Direct method: by using the definition,
Indirect method: by first evaluating the autocorrelation function and then taking the Fourier transform:
Pw(f)= ℑ [Rw(τ) ]
The average power can be obtained by any of the four techniques.

43. PSD of a Sinusoid

44. PSD of a Sinusoid
The average normalized power may be obtained by using:

45. Orthogonal Functions
Definition: Functions ϕn(t) and ϕm(t) are said to be Orthogonal with respect to each other the interval a < t < b if they satisfy the condition,
where
δnm is called the Kronecker delta function.
If the constants Kn are all equal to 1 then the ϕn(t) are said to be orthonormal functions.

46. Example 2.11 Orthogonal Complex Exponential Functions

47. 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 an 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 {φn} as a “Complete Set”
Examples for complete sets:
Harmonic Sinusoidal Sets {Sin(nw0t)}
Complex Expoents {ejnwt}
Bessel Functions
Legendare polynominals

48. Orthogonal Series
Proof of theorem: Assume that the set {φn} 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 to get,
Now, since we can find the coefficients an writing w(t) in series form is possible. Thus theorem is proved.

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

50. Ex. Square Waves Using Sine Waves.

51. Fourier Series Complex Fourier Series
The frequency f0 = 1/T0 is said to be the fundamental frequency and the frequency nf0 is said to be the nth harmonic frequency, when n>1.

52. Some Properties of Complex Fourier Series

53. Some Properties of Complex Fourier Series

54. Quadrature Fourier Series
The Quadrature Form of the Fourier series representing any physical waveform w(t) over the interval a < t < a+T0 is,
where the orthogonal functions are cos(nω0t) and sin(nω0t).
Using we can find the Fourier coefficients as:

55. Quadrature Fourier Series
Since these sinusoidal orthogonal functions are periodic, this series is periodic with the fundamental period T0.
The Complex Fourier Series, and the Quadrature Fourier Series are equivalent representations.
This can be shown by expressing the complex number cn as below
For all integer values of n
and
Thus we obtain the identities
and

56. Line Spectra for Periodic Waveforms
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.

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

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

59. Ex. 2.12 Fourier Coeff. for a Periodic Rectangular Wave
Now evaluate the coefficients from the Fourier Transform
 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.

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

61. END