Continuous-Time Signal Analysis FOURIER Transform - Applications DR. SIGIT PW JAROT ECE 2221

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1 Continuous-Time Signal Analysis FOURIER Transform - Applications DR. SIGIT PW JAROT ECE 2221

2 Inspiring Message from Imam Shafii You will not acquire knowledge unless you have 6 (SIX) THINGS Intelligence Strong Will Diligence Patience Sufficient Means 2 ECE 2221 Signals and Systems Befriend Your Teacher Dr. Sigit PW Jarot

To expose students to both the time-domain and frequencydomain methods of analyzing signals and systems.

3 Course Objectives To provide an analysis of the continuous-time signals and systems as reflected to their roles in engineering practice. To expose students to both the time-domain and frequencydomain methods of analyzing signals and systems. To illustrate the potential applications of this course as a Pre-requisite course to communication engineering and principles, digital signal processing and control system. 3 ECE 2221 Signals and Systems Dr. Sigit PW Jarot

4 OBE (Outcome Based Education) Learning Outcomes After completion of this course the students will be able to: Acquire intuitive and heuristic understanding of the concepts of signals and systems, and the physical meaning of the mathematical representation. Analyze continuous-time signals and systems in time domain. Analyze continuous-time signals and systems in frequency domain. 4 ECE 2221 Signals and Systems Acquire introductory-level knowledge of discrete-time signals and systems, and sampling theory. Dr. Sigit PW Jarot

5 5 Course Synopsis Introduction to Signals Introduction to Systems Time-Domain Analysis of Continuous-Time Systems Frequency-Domain System Analysis: the Laplace Transform MID-TERM Examination Signals Analysis using the Fourier Series Signals Analysis using the Fourier Transform Introduction to Discrete Time Signals and Systems Analysis FINAL Examination ECE 2221 Signals and Systems Dr. Sigit PW Jarot

6 Signals and Systems Input Signal System Output Signal SIGNAL A set of data or information. A function or sequence of values that represents information. A function of one or more variables (e.g. time, frequency, space,..) that conveys information on the nature of a physical phenomenon. SYSTEM A system is an entity that processes a set of signals (inputs) to yield another sets of signals (outputs) 6 ECE 2221 Signals and Systems Dr. Sigit PW Jarot

7 Outline Basic [ ] Motivation for using Fourier Transform Aperiodic Signal Representation by Fourier Integral Transforms of Some useful Functions More about FT [ ] Some Properties of the Fourier Transform Signal Transmission through LTI Systems Applications [ ] Ideal and Practical Filters Application to Communications Windowing

8 Partitioning a Complex Problem into Simpler Problems A common engineering technique is the partitioning of complex problems into simpler ones. The simpler problem are then solved, and the total solution becomes the sum of the simpler solutions. We may have insight into the simpler problems and can thus gain insight into the complex problems. 8 ECE2221 Signals and Systems DR Sigit Jarot

9 Three requirements 1. The problem can be expressed as a number of simpler problems. 2. The problem must be linear, such that the solution for the sum of function is equal to the sum of the solutions considering only one function at a time. 3. The contributions of the simpler solutions to the total solution must become negligible after considering a few terms. adequate acuracy 9 ECE2221 Signals and Systems DR Sigit Jarot

10 As engineers we must never lose sight of the fact that the mathematics that we employ is a means to an end and is not the end itself. Engineers apply mathematical procedures to the analysis and design of physical systems. 10 ECE2221 Signals and Systems DR Sigit Jarot

11 Why we need frequency domain representation? In many cases it is much easier to analyze the frequency content of a signal. 11 ECE2221 Signals and Systems DR Sigit Jarot

12 Why bother with the Fourier transform? There are certain simple time functions which are more readily represented by Fourier transforms than by Laplace transforms, e.g., x(t) = 1, x(t) = cos(2πft), periodic time functions, etc. Certain important operations on signals are more readily analyzed with Fourier transforms, e.g., sampling, modulation, filtering. Examination of both signals and systems in the frequency domain gives insights that complement those obtained in the time domain.

13 Fourier Transform Table (1)

14 Fourier Transform Table (2)

15 Fourier Transform Table (3)

16 Linearity Property 16 ECE 2221 Signals and Systems Dr. Sigit PW Jarot

17 Time-Frequency Duality of Fourier Transform 17 ECE 2221 Signals and Systems Dr. Sigit PW Jarot

18 Duality Property 18 ECE 2221 Signals and Systems Dr. Sigit PW Jarot

19 Duality Property: Example 19 ECE 2221 Signals and Systems Dr. Sigit PW Jarot

20 Duality Property Example Consider the FT of a rectangular function:

21 Scaling Property 21 ECE 2221 Signals and Systems Dr. Sigit PW Jarot

22 Time-shifting Property 22 ECE 2221 Signals and Systems Dr. Sigit PW Jarot

23 Time-shifting Property: Example Find the Fourier transform of the gate pulse x(t) given by: This pulse is rect(t/τ) delayed by 3τ/4 sec. Use time-shifting theorem, we get 23 ECE 2221 Signals and Systems Dr. Sigit PW Jarot

24 Frequency-shifting Property 24 ECE 2221 Signals and Systems Dr. Sigit PW Jarot

25 Frequency-Shifting Property : example Find and sketch the Fourier transform of the signal x(t)cos10t, where x(t)=rect(t/4) 25 ECE 2221 Signals and Systems Dr. Sigit PW Jarot

26 Convolution Property 26 ECE 2221 Signals and Systems Dr. Sigit PW Jarot

27 Convolution Property: proof 27 ECE 2221 Signals and Systems Dr. Sigit PW Jarot

28 Convolution Property: example Find and sketch the Fourier transform of the signal x(t)cos10t, where x(t)=rect(t/4), using convolution property Dr. Sigit PW Jarot ECE 2221 Signals and Systems 28

29 Time-differentiation Property 29 ECE 2221 Signals and Systems Dr. Sigit PW Jarot

30 30 ECE2221 Signals and Systems DR Sigit Jarot

31 Signal Transmission through LTI Systems We have seen previously that if x(t) and y(t) are input & output of a LTI system with impulse response h(t), then Y(ω)=H(ω) X(ω) We can therefore perform LTI system analysis with Fourier transform in a way similar to that of Laplace transform. However, FT is more restrictive than Laplace transform because the system must be stable, and x(t) must itself by Fourier transformable. Laplace transform can be used to analyse stable AND unstable system, and apply to signals that grow exponentially. If a system is stable, it can shown that the frequency response of the system H(jω) is just the Fourier transform of h(t) (i.e. H(ω)): H(ω)=H(s) s=j ω 31 ECE 2221 Signals and Systems Dr. Sigit PW Jarot

32 Example 32 ECE 2221 Signals and Systems Dr. Sigit PW Jarot

33 Time-domain vs. Frequency-domain 33 ECE 2221 Signals and Systems Dr. Sigit PW Jarot

34 Signal Distortion during transmission QUESTION: What is the characteristic of a system that allows signal to pass without distortion? Transmission is distortionless if output is identical to input within a multiplicative constant, and relative delay is allowed. That is: But Y(ω)/X(ω) = H(ω), therefore the frequency characteristic of a distortionless system is: For distortionless transmission, amplitude response H(ω) must be a constant AND phase response H(ω) must be linear function of ω with slope t d. 34 ECE 2221 Signals and Systems Dr. Sigit PW Jarot

35 35 ECE 2221 Signals and Systems Dr. Sigit PW Jarot

36 Parseval s Theorem 36 ECE 2221 Signals and Systems Dr. Sigit PW Jarot

37 Energy Spectral Density of a Signal 37 ECE 2221 Signals and Systems Dr. Sigit PW Jarot

38 If x(t) is a real signal, then X(ω) and X(-ω) are conjugate : This implies that X(ω) is an even function. Therefore Consequently, the energy contributed by a real signal by spectral components between ω 1 and ω 2 is: 38 ECE 2221 Signals and Systems Dr. Sigit PW Jarot

39 Example: Find the energy E of signal x(t) = e -at u(t). Determine the frequency W (rad/s) so that the energy contributed by the spectral component from 0 to W is 95% of the total signal energy E. 39 ECE 2221 Signals and Systems Dr. Sigit PW Jarot

40 Filter A filter separates the wanted part of a signal from the useless part In the signal and system analysis, it is a device which separates the signal in one frequency range from the signal in another frequency range An ideal filter passes all signal power in its passband without distortion and completely blocks signal power outside its passband 40 ECE2221 Signals and Systems DR Sigit Jarot

41 Filter Has transfer function, H( ) that allows/passed frequency component of input signal within the passband and eliminate the frequency component of input signal within the stopband 41 ECE2221 Signals and Systems DR Sigit Jarot

42 Ideal Filters Has transfer function, H( ) that pass the frequency component of input signal within the passband and eliminate those outside it. 42 ECE2221 Signals and Systems DR Sigit Jarot

43 Ideal Filters PASSBAND = unity magnitude frequency response STOPBAND = zero frequency response 43 ECE2221 Signals and Systems DR Sigit Jarot

44 Ideal Filters The output of the filter consist only the frequency components of input signal that are within the passband 44 ECE2221 Signals and Systems DR Sigit Jarot

45 Ideal Low-pass Filters H(j ) - C 0 C Stopband Passband Stopband 45 ECE2221 Signals and Systems DR Sigit Jarot

46 Ideal High-pass Filters H(j ) - C 0 C Passband Stopband Passband 46 ECE2221 Signals and Systems DR Sigit Jarot

47 Ideal Bandpass Filters H(j ) StopbandPassbandStopbandPassbandStopband 47 ECE2221 Signals and Systems DR Sigit Jarot

48 Ideal Bandstop Filters H(j ) PassbandStopbandPassbandStopbandPassband 48 ECE2221 Signals and Systems DR Sigit Jarot

49 Example Classify each of these transfer functions as having a lowpass, highpass, bandpass or bandstop frequency response Lowpass 49 ECE2221 Signals and Systems DR Sigit Jarot

50 Example Bandpass Bandstop 50 ECE2221 Signals and Systems DR Sigit Jarot

51 Example Bandpass Bandpass 51 ECE2221 Signals and Systems DR Sigit Jarot

52 Example Bandstop 52 ECE2221 Signals and Systems DR Sigit Jarot

53 H Ideal Lowpass Filter C ( ) rect F -1 h( t) C sinc 2 C 1 t C 0 C Frequency response Impulse response The impulse response for this ideal filter implies a noncausal system, it begins long before the impulse occurs at t = 0 t 53 ECE2221 Signals and Systems DR Sigit Jarot

54 Impulse Response and Causality All the impulse responses of ideal filters are sinc functions, or related functions, which are infinite in extent Therefore all ideal filter impulse responses begin before time, t = 0 This makes ideal filters non-causal Ideal filters are not physically realizable 54 ECE2221 Signals and Systems DR Sigit Jarot

55 Impulse Responses and Frequency Responses of Real Causal Filters 55 ECE2221 Signals and Systems DR Sigit Jarot

56 Impulse Responses and Frequency Responses of Real Causal Filters 56 ECE2221 Signals and Systems DR Sigit Jarot

57 Real Filters-RC Lowpass Filter H j V out j j Z c V in Z c j j Z R j 1 j RC 1 57 ECE2221 Signals and Systems DR Sigit Jarot

58 DR Sigit Jarot ECE2221 Signals and Systems 58

59 Time Domain vs. Frequency Domain x (t) = rect (t) F x (f) = sinc (f) t f Time duration/ Time interval Bandwidth 59 ECE2221 Signals and Systems DR Sigit Jarot

60 Absolute Bandwidth X(f) -f 2 -f 1 f 1 B f 2 f DR Sigit Jarot ECE2221 Signals and Systems 60

61 3dB (or Half-Power) Bandwidth H(f 0 ) H(f 0 ) 2 f -f 2 -f 1 f 1 f 2 B H(f ) H(f 0 ) -f 1 0 f 1 2 f DR Sigit Jarot ECE2221 Signals and Systems 61

62 Null-to-null (or Zero Crossing) Bandwidth X(f) -f 2 -f 1 f 1 f 2 f B 0 DR Sigit Jarot ECE2221 Signals and Systems 62 B f 1 f

E C E S I G N A L S A N D S Y S T E M S. ECE 2221 Signals and Systems, Sem /2011, Dr. Sigit Jarot

E C E S I G N A L S A N D S Y S T E M S. ECE 2221 Signals and Systems, Sem /2011, Dr. Sigit Jarot 1 E C E 2 2 2 1 S I G N A L S A N D S Y S T E M S ECE 2221 Signals and Systems, Sem 3 2010/2011, Dr. Sigit Jarot Outline Course Objectives Learning Outcomes Course Synopsis Text and Supporting Books Course