1. Chapter 3 Fourier Representation of Signals
Ya Bao

2. Signal representations
Sinusoidal signal
Note: ω=2πf
Or, or
Where: A is the sinusoid's amplitude ω is the angular velocity of the sinusoid in radian/s, θ is an arbitrary phase in radian.
Ya Bao

3. Time domain graph
Ya Bao

4. Frequency domain spectra
Ya Bao

5. Classification of signals
Energy signals, Power signals
An energy signal is a pulse-like signal that usually exits for only a finite interval of time or has a major portion of its energy concentrated in a finite time interval.
Ya Bao

6. Classification of signals (cont1)
An energy signal is defined to be one for which the
Is finite even when the time interval becomes infinite; i.e., when
Average power dissipated by the signal f(t) during the time interval (t1, t2) is
Ya Bao

7. Classification of signals (cont2)
Power signal
Then the signal f(t) has finite but nonzero average power when the time interval becomes infinite, it is called a power signal.
Ya Bao

8. Classification of signals (cont3)
Periodic, Nonperiodic (aperiodic)
A periodic signal is one that repeats itself exactly after a fixed length of time.
T – period, it define the duration of one complete cycle of f(t)
If its energy per cycle is finite then it is power signal. The average power need only be calculated over one complete cycle.
Any signal for which there is no value of period T is said nonperiodic (or aperiodic) signal.
Ya Bao

9. Classification of signals (cont4)
Deterministic, non-deterministic (random)
Deterministic signal: no uncertainty in its values.
an explicit mathematical expression can be written
Random signal: some degree of uncertainty before it actual occurs. (discussed later)
A collection of signals, each of which is different e.g. uncertain starting phase
Future values of the signal may not be predictable. E.g. noise
Ya Bao

10. Multiplication and Convolution
Multiplication in frequency domain is convolution in time domain.
Convolution in frequency domain is multiplication in time domain.
Convolution may be defined
Convolution is an alternative operation of multiplication.
Convolution and multiplication are complementary, but not identical.
Where g(u) is simply g(t) with t replaced by dummy variable u.
Ya Bao
By Ya Bao

11. Fourier Series for period signal
Jean Baptiste Joseph Fourier (21 March – 16 May 1830) , was a French mathematician and physic ist best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations.
Ya Bao

12. Fourier Series for period signal
Fourier proved that any period signal can be represented as a sum of sinusoids, known as Fourier series.
𝑠 𝑡 = 𝐴 ∞ 𝐴 𝑛 cos 2𝜋𝑛𝑓𝑡 + 1 ∞ 𝐵 𝑛 sin⁡(2𝜋𝑛𝑓𝑡)
A0 is the average value of the signal over a period.
An is the coefficient of the nth cosine component.
Bn is the coefficient of the nth sine component.
Ya Bao

13. f: fundamental frequency or fundamental harmonic;
A periodic signal with period T consists of the fundamental frequency f=1/T plus integer multiples of that frequency (harmonics).
If A0 ≠0, then s(t) has a dc component, average voltage is
The root mean square (rms) voltage is
Ya Bao

14. Example A0 = 0, no dc voltage Bn = 0, no sine components
Only odd harmonics.
Amplitudes are alternatively positive and negative and it approaches zero as we move toward infinity.
Ya Bao

15. Example
Ya Bao

16. Fourier Transform for nonperiodic signal.
While the Fourier series gives the discrete frequency domain of a period signal, the Fourier Transform gives the continuous domain of a nonperiodic signal.
Ya Bao

17. 2.1 The Fourier Transform Definitions
Fourier Transform of the signal g(t)
A lowercase letter to denote the time function and an uppercase letter to denote the corresponding frequency function
Infinite integrals; definite integrals
Eternal:
Ya Bao
Ya Bao

18. Ya Bao

19. Notations The frequency f is related to the angular frequency ω as
Shorthand notation for the transform relations
Ya Bao

20. Continuous Spectrum
A pulse signal g(t) of finite energy is expressed as a continuous sum of exponential functions with frequencies in the interval -∞ to ∞.
A signal can be defined by time domain function g(t), as well as defined by G(f).
The Fourier transform G(f) is a complex function of frequency f,
Ya Bao

21. The spectrum of a real-valued signal
For the special case of a real-valued function g(t)
The spectrum of a real-valued signal
The amplitude spectrum of the signal is an even function of the frequency, the amplitude spectrum is symmetric with respect to the origin f = 0.
The phase spectrum of the signal is an odd function of the frequency, the phase spectrum is antisymmetric with respect to the origin f = 0.
Ya Bao

22. X(t) = Acos(ω0t+θ)
Ya Bao

23. Ya Bao

24. Ya Bao

25. = =
Ya Bao

26. Inverse relationship between time-domain and frequency-domain descriptions of a signal. A pulse narrow in time has a significant frequency description over a wide range of frequencies, and vice versa.
The rectangular pulse g(t) is a symmetric function of time t; the Fourier transform G(f) is asymmetric function of frequency f.
Ya Bao

27. 2.2 Properties of the Fourier Transrom
Property 1 : Linearity (Superposition)
then for all constants c1 and c2,
Ya Bao

28. Property 2 : Dilation
The dilation factor (a) is real number
The compression of a function g(t) in the time domain is equivalent to the expansion of its Fourier transform G(f) in the frequency domain by the same factor, or vice versa.
Reflection property
For the special case when a=-1
Ya Bao

29. Property 3 : Conjugation Rule
then for a complex-valued time function g(t),
Where the asterisk denotes the complex-conjugate operation.
Property 4 : Duality
Ya Bao

30. Property 5 : Time Shifting
If a function g(t) is shifted along the time axis by an amount t0, the effect is equivalent to multiplying its Fourier transform G(f) by the factor exp(-j2πft0). This means the amplitude of G(f) is unaffected by the time shift, but its phase is changed by the linear factor – 2πft0
Property 6 : Frequency Shifting
A shift of the range of frequencies in a signal is accomplished by using the process of modulation.
Ya Bao

31. Property 7 : Area Under g(t)
The area under a function g(t) is equal to the value of its Fourier transform G(f) at f=0.
Property 8 : Area under G(f)
The value of a function g(t) at t=0 is equal to the area under its Fourier transform G(f).
Property 9 : Differentiation in the Time Domain
Differentiation of a time function g(t) has the effect of multiplying its Fourier transform G(f) by the purely imaginary factor j2πf.
Ya Bao

32. Ya Bao

33. Ya Bao

34. Property 10 : Integration in the Time Domain
Ya Bao

35. Modulation theorem Property 11 : Modulation Theorem
The multiplication of two signals in the time domain is transformed into the convolution of their individual Fourier transforms in the frequency domain.
Shorthand notation
Ya Bao

36. Property 12 : Convolution Theorem
Convolution of two signals in the time domain is transformed into the multiplication of their individual Fourier transforms in the frequency domain.
Property 13 : Correlation Theorem
The integral on the left-hand side of Eq.(2.53) defines a measure of the similarity that may exist between a pair of complex-valued signals
Ya Bao

37. Property 14 : Rayleigh’s Energy Theorem
Total energy of a Fourier-transformable signal equals the total area under the curve of squared amplitude spectrum of this signal.
Ya Bao

38. Ya Bao

39. 2.3 The Inverse Relationship Between Time and Frequency
If the time-domain description of a signal is changed, the frequency-domain description of the signal is changed in an inverse manner, and vice versa.
If a signal is strictly limited in frequency, the time- domain description of the signal will trail on indefinitely, even though its amplitude may assume a progressively smaller value. – a signal cannot be strictly limited in both time and frequency.
Bandwidth
A measure of extent of the significant spectral content of the signal for positive frequencies.
Ya Bao

40. Commonly used three definitions
Null-to-null bandwidth
When the spectrum of a signal is symmetric with a main lobe bounded by well-defined nulls – we may use the main lobe as the basis for defining the bandwidth of the signal
3-dB bandwidth
Low-pass type : The separation between zero frequency and the positive frequency at which the amplitude spectrum drops to 1/√2 (0.707) of its peak value.
Band-pass type : the separation between the two frequencies at which the amplitude spectrum of the signal drops to 1/√2 of the peak value at fc.
Root mean-square (rms) bandwidth
The square root of the second moment of a properly normalized form of the squared amplitude spectrum of the signal about a suitably chosen point.
Ya Bao

41. Ya Bao

42. Time-Bandwidth Product
The produce of the signal’s duration and its bandwidth is always a constant
Whatever definition we use for the bandwidth of a signal, the time-bandwidth product remains constant over certain classes of pulse signals
the corresponding definition for the rms duration of the signal g(t) is
The time-bandwidth product has the following form
Ya Bao

43. Delta function
Having zero amplitude everywhere except at t=0, where it is infinitely large in such a way that it contains unit area under its curve.
We may express the integral of the product g(t)δ(t-t0) with respect to time t as follows :
The convolution of any time function g(t) with the delta function δ(t) leaves that function completely unchanged. – replication property of delta function.
The Fourier transform of the delta function is
Shifting property of the delta function
Ya Bao

44. The Fourier transform pair for the Delta function
The delta function as the limiting form of a pulse of unit area as the duration of the pulse approaches zero.
A rather intuitive treatment of the function along the lines described herein often gives the correct answer.
Ya Bao

45. Ya Bao

46. Applications of the Delta Function
DC signal
By applying the duality property to the Fourier transform pair of Eq.(2.65)
A dc signal is transformed in the frequency domain into a delta function occurring at zero frequency
Ya Bao

47. Sinusoidal Functions The Fourier transform of the cosine function
The spectrum of the cosine function consists of a pair of delta functions occurring at f=±fc, each of which is weighted by the factor ½

48. Unit Step Function
The unit step function u(t) equals +1 for positive time and zero for negative time.
The unit step function and signum function are related by
Unit step function is represented by the Fourier-transform pair
The spectrum of the unit step function contains a delta function weighted by a factor of ½ and occurring at zero frequency
Ya Bao

49. Theme Example : Twisted Pairs for Telephony
The typical response of a twisted pair with lengths of 2 to 8 kilometers
Twisted pairs run directly from the central office to the home with one pair dedicated to each telephone line. Consequently, the transmission lines can be quite long
The results in Fig. assume a continuous cable. In practice, there may be several splices in the cable, different gauge cables along different parts of the path, and so on. These discontinuities in the transmission medium will further affect the frequency response of the cable.
Ya Bao
Ya Bao

50. Online tutorials (strongly recommended)
We see that, for a 2-km cable, the frequency response is quite flat over the voice band from 300 to 3100 Hz for telephonic communication. However, for 8-km cable, the frequency response starts to fall just above 1 kHz.
Online tutorials (strongly recommended)
Fourier Series Tutorial
Tutorial based on a combination of audio lectures and interactive flash modules. An inovative way to learn this difficult subject.
Ya Bao

51. End !
Ya Bao