Transcription of Fourier Series and Fourier Transform
1 Spring 2007 Fourier Series and Fourier Transform , Slide 1 Fourier Seriesand Fourier Transform Complex exponentials Complex version of Fourier Series Time Shifting, Magnitude, Phase Fourier TransformCopyright 2007 by PerrottAll rights reserved. Spring 2007 Fourier Series and Fourier Transform , Slide 2 The Complex Exponential as a Vector Euler s Identity:Note: Consider Iand Qas the realand imaginaryparts As explained later, in communication systems, Istands for in-phaseand Qfor quadrature As tincreases, vector rotates counterclockwise We consider ejwtto have positivefrequencye j tIQcos( t)sin( t) Spring 2007 Fourier Series and Fourier Transform , Slide 3 The Concept of Negative FrequencyNote: As tincreases, vector rotates clockwise We consider e-jwtto have negativefrequency Note: A-jBis the complex conjugateof A+jB So,e-jwtis the complex conjugate of ejwte-j tIQcos( t)-sin( t) Spring 2007 Fourier Series and Fourier Transform , Slide 4 Add Positive and Negative FrequenciesNote.
2 As tincreases, the additionof positiveand negativefrequency complex exponentials leads to a cosinewave Note that the resulting cosine wave is purely realand considered to have a positivefrequencye-j tIQej t2cos( t) Spring 2007 Fourier Series and Fourier Transform , Slide 5 Subtract Positive and Negative FrequenciesNote: As tincreases, the subtractionof positiveand negativefrequency complex exponentials leads to a sinewave Note that the resulting sine wave is purely imaginaryand considered to have a positivefrequency-e-j tIQej t2sin( t) Spring 2007 Fourier Series and Fourier Transform , Slide 6 Fourier Series The Fourier Series is compactly defined using complex exponentials Where:tTx(t) Spring 2007 Fourier Series and Fourier Transform , Slide 7 From The Previous Lecture The Fourier Series can also be written in terms of cosines and sines:tTx(t) Spring 2007 Fourier Series and Fourier Transform , Slide 8 Compare Fourier Definitions Let us assume the following: Then: Spring 2007 Fourier Series and Fourier Transform , Slide 9 Square Wave ExampletTT/2x(t) Spring 2007 Fourier Series and Fourier Transform , Slide 10 Graphical View of Fourier Series As in previous lecture, we can plot Fourier Series coefficients Note that we now have positiveand negativevalues of n Square wave example.
3 2A 2A3 -2A -2A3 nnBnAn13579-9 -7 -5 -3 -113579-9 -7 -5 -3 Spring 2007 Fourier Series and Fourier Transform , Slide 112A 2A3 -2A -2A3 ffBfAf1T-3T-5T-1T3T5T7T9T-7T-9T1T-3T-5T- 1T3T5T7T9T-7T-9 TIndexing in Frequency A given Fourier coefficient, ,represents the weight corresponding to frequency nwo It is often convenient to index in frequency (Hz) Spring 2007 Fourier Series and Fourier Transform , Slide 12 The Impact of a Time (Phase) Shift Consider shifting a signal x(t)in time by TdtT/4x(t)TT/4A-AtTT/2x(t)A-A Define: Which leads Spring 2007 Fourier Series and Fourier Transform , Slide 13 Square Wave Example of Time Shift To simplify, note that exceptfor oddntT/4x(t)TT/4A-AtTT/2x(t) Spring 2007 Fourier Series and Fourier Transform , Slide 142A ffBfAf1T-3T-5T-1T3T5T7T9T-7T-9T1T-3T-5T- 1T3T5T7T9T-7T-9T2A 2A3 -2A -2A3 ffBfAf1T-3T-5T-1T3T5T7T9T-7T-9T1T-3T-5T- 1T3T5T7T9T-7T-9 TGraphical View of Fourier SeriestT/4x(t)TT/4A-AtTT/2x(t) Spring 2007 Fourier Series and Fourier Transform , Slide 15 Magnitude and Phase We often want to ignore the issue of time (phase) shifts when using Fourier analysis Unfortunately, we have seen that the Anand Bncoefficients are very sensitive to time (phase)
4 Shifts The Fourier coefficients can also be represented in term of magnitude and phase Spring 2007 Fourier Series and Fourier Transform , Slide 16 Graphical View of Magnitude and PhasetT/4x(t)TT/4A-AtTT/2x(t)A-Af-3T-5T- 1T-7T-9T2A 2A3 ffXf f1T-3T-5T-1T3T5T7T9T-7T-9T1T-3T-5T-1T3T5 T7T9T-7T-9T2A 2A3 f2A 2A3 fXf1T-3T-5T-1T3T5T7T9T-7T-9T2A 2A3 - /2 /21T3T5T7T9T Spring 2007 Fourier Series and Fourier Transform , Slide 17 Does Time Shifting Impact Magnitude? Consider a waveform x(t)along with its Fourier Series We showed that the impact of time (phase) shifting x(t)on its Fourier Series is We therefore see that time (phase) shifting does notimpact the Fourier Series Spring 2007 Fourier Series and Fourier Transform , Slide 18 Parseval s Theorem The squared magnitude of the Fourier Series coefficients indicates power at corresponding frequencies Power is defined as:Note:* Spring 2007 Fourier Series and Fourier Transform , Slide 19 The Fourier Transform The Fourier Series deals with periodicsignals The Fourier Transform deals with Spring 2007 Fourier Series and Fourier Transform , Slide 20 Fourier Transform Example Note that x(t)is notperiodic Calculation of Fourier Transform .
5 Tx(t) Spring 2007 Fourier Series and Fourier Transform , Slide 21X(j2 f)2TA2T12T-1fGraphical View of Fourier Transformtx(t)TA-TThis is calleda Spring 2007 Fourier Series and Fourier Transform , Slide 22 Summary The Fourier Series can be formulated in terms of complex exponentials Allows convenient mathematical form Introduces concept of positive and negative frequencies The Fourier Series coefficients can be expressed in terms of magnitude and phase Magnitude is independent of time (phase) shifts of x(t) The magnitude squared of a given Fourier Series coefficient corresponds to the power present at the corresponding frequency The Fourier Transform was briefly introduced Will be used to explain modulation and filtering in the upcoming lectures We will provide an intuitive comparison of Fourier Series and Fourier Transform in a few weeks.
