Department of Electrical Engineering University of Arkansas ELEG 3124 SYSTEMS AND SIGNALS Ch. 5 Fourier Transform Dr. Jingxian Wu wuj@uark.edu
OUTLINE 2 Introduction Fourier Transform Properties of Fourier Transform Applications of Fourier Transform
INTRODUCTION: MOTIVATION 3 Motivation: Fourier series: periodic signals can be decomposed as the summation of orthogonal complex exponential signals x( cn exp jn0t 1 T cn x( exp jn0t n T 0 each harmonic contains a unique frequency: n/t dt time domain frequency domain How about aperiodic signals T?
INTRODUCTION: TRANSFER FUNCTION 4 System transfer function j t e ) h(t e jt H() H( ) h( exp System with periodic inputs e 0 jn t h( e jt dt H( n 0) jn 0 t jn0t cne n h( n c n e H( n ) 0 jn 0 t x( h( cn n e H( n ) 0 jn 0 t
OUTLINE 5 Introduction Fourier Transform Properties of Fourier Transform Applications of Fourier Transform
FOURIER TRANSFORM Fourier Transform 6 X ( ) x( e given x(, we can find its Fourier transform Inverse Fourier Transform x( 1 2 X ( ) e jt given X (), we can find the time domain signal x( signal is decomposed into the weighted summation of complex exponential functions. (integration is the extreme case of summation) dt j t d x ( X () X ()
FOURIER TRANSFORM 7 Example x( rect ( t / Find the Fourier transform of )
FOURIER TRANSFORM 8 Example Find the Fourier transform of x( exp( a t ) a 0
FOURIER TRANSFORM 9 Example Find the Fourier transform of x( exp( a u( a 0
FOURIER TRANSFORM 10 Example Find the Fourier transform of x( ( t a)
FOURIER TRANSFORM: TABLE 11 Department of Engineering Science Sonoma State University
FOURIER TRANSFORM The existence of Fourier transform Not all signals have Fourier transform If a signal have Fourier transform, it must satisfy the following two conditions Example 1. x( is absolutely integrable x ( dt x( exp( u( 2. x( is well behaved The signal has finite number of discontinuities, minima, and maxima within any finite interval of time. 12
OUTLINE 13 Introduction Fourier Transform Properties of Fourier Transform Applications of Fourier Transform
PROPERTIES: LINEARITY Linearity If x X ( ) x X ( ) 1( 1 2( 2 14 then ax ( bx2( ax1( ) bx 2( 1 ) Example Find the Fourier transform of x( 2rect ( t / ) 3exp( 2 u( 4 (
PROPERTY: TIME-SHIFT 15 Time shift If x( X ( ) Then x( t t0) X ( )exp[ jt0 ] phase shift Review: complex number j c c e c cos( ) j c sin( ) a a c cos c b b c sin a tan( b / a 2 2 a ) Phase shift of a complex number c by : 0 time shift in time domain frequency shift in frequency domain jb c j ) c exp j( ) exp( 0 0
PROPERTY: TIME SHIFT 16 Example: Find the Fourier transform of x( rect t 2
PROPERTY: TIME SCALING 17 Time scaling If x( X ( ) Then x( a 1 a X a Example Let X ( ) rect 1 / 2, find the Fourier transform of x( 2t 4)
PROPERTY: SYMMETRY 18 Symmetry x( If x( X ( ), and is a real-valued time signal Then X ( ) X * ( )
PROPERTY: DIFFERENTIATION 19 Differentiation If x( X ( ) Then dx( dt ) jx ( ) n d x( n dt n j X ( Example Let X ( ) rect 1 / 2, find the Fourier transform of dx ( dt
PROPERTY: DIFFERENTIATION 20 Example Find the Fourier transform of x( sgn( d 1 (Hint: sgn( ( ) dt 2
PROPERTY: CONVOLUTION 21 Convolution If x( X ( ), h( H( ) Then x( h( X ( ) H( ) x( h( x( h( X () H() X ( ) H( ) time domain frequency domain
PROPERTY: CONVOLUTION 22 Example An LTI system has impulse response h( exp at u( ) If the input is x( ( a b)exp bt u( ( c a)exp( c u( t Find the output ( a 0, b 0, c 0)
PROPERTY: MULTIPLICATION 23 Multiplication If x( X ( ), m( M( ) Then 1 x( m( X ( ) M ( ) 2
PROPERTY: DUALITY 24 Duality If g( G( ) Then G( 2g ( )
PROPERTY: DUALITY 25 Example Find the Fourier transform of h( Sa t 2 (recall: rect ( t / ) sinc ) 2
PROPERTY: DUALITY 26 Example Find the Fourier transform of x( 1 Find the Fourier transform of x 0 ( e j t
PROPERTY: SUMMARY 27
PROPERTY: EXAMPLES 28 Examples 1. Find the Fourier transform of x ( t ) cos( t 0 ) 2. Find the Fourier transform of x( u( 1 2 u( sgn( 1 sgn( 2 j
PROPERTY: EXAMPLES 29 Examples 3. A LTI system with impulse response Find the output when input is at u( ) h( exp t x( u( 4. If x( X ( ), find the Fourier transform of t x( ) d t (Hint: x( ) d x( u( )
PROPERTY: EXAMPLES 30 Example 5. (Modulation) If x( X ( ), m ( t ) cos( t 0 ) Find the Fourier transform of x( m( 6. If 1 X ( ), find x( a j
PROPERTY: DIFFERENTIATION IN FREQ. DOMAIN 31 Differentiation in frequency domain If: Then: x( X ( ) n ( j x( n d X () n d
PROPERTY: DIFFERENTIATION IN FREQ. DOMAIN 32 Example Find the Fourier transform of t exp( a u(, a 0
PROPERTY: FREQUENCY SHIFT 33 Frequency shift If: Then: x( X ( ) x( exp( j0 X ( 0) Example If X ( ) rect 1 / 2, find the Fourier transform x( exp( j2
PROPERTY: PARSAVAL S THEOREM 34 Review: signal energy E Parsaval s theorem x( 2 dt x( 1 2 2 dt X ( ) 2 d
PROPERTY: PARSAVAL S THEOREM 35 Example: Find the energy of the signal x( exp( 2 u(
PROPERTY: PERIODIC SIGNAL 36 Fourier transform of periodic signal Periodic signal can be written as Fourier series n x( c exp jn t n 0 Perform Fourier transform on both sides n X ( ) 2 cn ( n0)
OUTLINE 37 Introduction Fourier Transform Properties of Fourier Transform Applications of Fourier Transform
APPLICATIONS: FILTERING 38 Filtering Filtering is the process by which the essential and useful part of a signal is separated from undesirable components. Passing a signal through a filter (system). At the output of the filter, some undesired part of the signal (e.g. noise) is removed. Based on the convolution property, we can design filter that only allow signal within a certain frequency range to pass through. x( h( x( h( X () H() X ( ) H( ) filter filter time domain frequency domain
APPLICATIONS: FILTERING 39 Classifications of filters Low pass filter High pass filter Band pass filter Band stop (Notch) filter
APPLICATIONS: FILTERING 40 Practical filters (non-ideal filters)
APPLICATION: FILTERING 41 A filtering example A demo of a notch filter X () H() X ( ) H( )
APPLICATIONS: FILTERING 42 Example Find out the frequency response of the RC circuit What kind of filters it is?
APPLICATION: SAMPLING THEOREM 43 Sampling theorem: time domain Sampling: convert the continuous-time signal to discrete-time signal. x( n p ( ( t nt) sampling period x s ( x( p(
APPLICATION: SAMPLING THEOREM 44 Sampling theorem: frequency domain Fourier transform of the impulse train impulse train is periodic p( n Fourier series 1 jnst ( t nts ) 1e T s n Find Fourier transform on both sides 2 P ( ) ( ns ) T s n Time domain multiplication Frequency domain convolution 1 x( p( X ( ) P( ) 2 1 x ( p( X ( ns ) T s n s 2 T s
APPLICATION: SAMPLING THEOREM 45 Sampling theorem: frequency domain Sampling in time domain Repetition in frequency domain
APPLICATION: SAMPLING THEOREM 46 Sampling theorem If the sampling rate is twice of the bandwidth, then the original signal can be perfectly reconstructed from the samples. 2 s B s 2 B s 2 B Department of Engineering Science Sonoma State University s 2 B
APPLICATION: AMPLITUDE MODULATION 47 What is modulation? The process by which some characteristic of a carrier wave is varied in accordance with an information-bearing signal Information bearing signal Three signals: modulation Carrier wave Information bearing signal (modulating signal) Usually at low frequency (baseband) E.g. speech signal: 20Hz 20KHz Carrier wave Usually a high frequency sinusoidal (passband) Modulated signal E.g. AM radio station (1050KHz) FM radio station (100.1MHz), 2.4GHz, etc. Modulated signal: passband signal
APPLICATION: AMPLITUDE MODULATION 48 Amplitude Modulation (AM) s( A m( cos(2 f c A direct product between message signal and carrier signal c m( Mixer s( A c Local Oscillator cos( 2f c
APPLICATION: AMPLITUDE MODULATION 49 Amplitude Modulation (AM) Ac S( f ) M ( f fc) M ( f fc) 2