Signals and Systems Continuous-Time Fourier Transform Chang-Su Kim continuous time discrete time periodic (series) CTFS DTFS aperiodic (transform) CTFT DTFT
Lowpass Filtering Blurring or Smoothing Original average Strong LPF e.g. 21x21 moving average filter Less strong LPF e.g. 11x11 moving average filter
Highpass Filtering Edge Extraction
CTFT Formula and Its Derivation
Bridge Between Fourier Series and Transform Consider the periodic signal x(t) x(t) -T -T 1 0 T 1 T t Its Fourier coefficients are a 2T1, T sin( 0T 1 ) 0 0
Bridge Between Fourier Series and Transform Setch a on the -axis a 2T 1 /T -2-1 0 1 2 The setch is obtained by sampling the sinc function. For each value of, the signal x(t) has a periodic component with weight a. So, the above setch shows the frequency content of the signal x(t).
Bridge Between Fourier Series and Transform The same setch a on the -axis: a 2T 1 /T -2 0-0 0 0 2 0 On the -axis, the distance between two consecutive a s is 0 =2/T, which is the fundamental frequency.
Bridge Between Fourier Series and Transform The same setch Ta on the -axis: Ta 2T 1 X(jw) -2 0-0 0 0 2 0 The distance between two adjacent a s is 0 =2/T. As T, 0 0. The distance between two consecutive a s becomes zero The setch of a becomes continuous The continuous curve X(jw) is called as Fourier Transform
Bridge Between Fourier Series and Transform On the other hand, as T, the signal x(t) becomes an aperiodic signal x(t) -T 1 0 T 1 t Fourier Transform can represent an aperiodic signal in frequency domain
From CTFS to CTFT: Formal Derivation How can we use this formula for a nonperiodic (aperiodic) function x(t)? x(t) 0 ~ x ( t )... 0 T 2T x( t) lim x( t) T
From CTFS to CTFT: Formal Derivation Given the relationships x t j0t ( ) a e, 1 T j0t a x() t e dt T x( t) lim x( t) T derive the following CTFT formula 1 jt x( t) X ( j) e d 2 jt X ( j) x( t) e dt
CTFT Formula Fourier Transform Pair Forward Transform jt X ( j) x( t) e dt Inverse Transform 1 jt x( t) X ( j) e d 2 X(jw) represents the strength of frequency component at w in x(t)
Time Domain vs. Frequency Domain Fourier analysis (series or transform) is a tool to determine the frequency contents of a given signal Conversion from time domain to frequency domain. jt X ( j) x( t) e dt It is always possible to move bac from frequency domain to time domain 1 jt x( t) X ( j) e d 2
Some Examples Ex 1) Impulse function constant function 1 0 t x( t) ( t) F X ( j) ( t) e jt dt 1
Some Examples Ex 2) Rectangular pulse sinc function 1 -T 1 T 1 t T 1 T 2sinT T sin( t) where sinc( t) t 1 j t 1 1 ( j) e dt 2 T1 sinc( ) T1 2T 1 T 1 Note the inverse relationship between time and frequency domains
More Examples
Unified Framewor for CTFS and CTFT: Periodic Signals Can Also Be Represented as Fourier Transform
Fourier Transform for Periodic Signals Consider the inverse Fourier transform of ( ) 2a ( 0 j ) So, we can deduce that j0t Fourier ( ) ( ) 2 ( 0) x t a e j a... 0 2 0 3 0... 2a 3
Fourier Transform for Periodic Signals Ex 1) sin function Ex 2) cos function F. S. 1 x( t) sin( 0t) a1, a 1 2 j 1 2 j F. S. x( t) cos( t) a1 a 1 0 1 2 ( j) ( j) /j 0 0 0 0 -/j
Fourier Transform for Periodic Signals Ex 3) Fourier transform of impulse trains 1 T /2 1 FS.. j0t x( t) ( t T ) a ( t) e dt T T /2 T 2 2 X( j) ( ) T T x(t) X( j) 1 F 2/T... T 2T t 2/T 4/T
Properties of CTFT
Properties of CTFT 1. Linearity F a x( t) b y( t) ax ( j) by ( j) 2. Time shifting x t t e X j F jt0 ( ) ( ) 0 3. Conjugation and conjugate symmetry * F * x t X j ( ) ( ) X j X j x t * ( ) ( ) [ ( ) real] 4. Differentiation and integration dx() t F j ( j) dt t F 1 x( ) d X ( j) X (0) ( ) j
Properties of CTFT 5. Time and frequency scaling 6. Parseval s relation 7. Duality ) ( ) ( ) ( 1 ) ( j t x a j a at x F F d j dt t x 2 2 ) ( 2 1 ) ( ) ( 2 ) ( ) ( ) ( F F g jt G j G t g
Convolution Property of CTFT F y( t) h( t) x( t) Y ( j ) H( j) X ( j) Two approaches for proof and understanding 1. LTI interpretation Note that the frequency response H(jw) is just the CTFT of the impulse response h(t). 2. Direct equation manipulation
Convolution Property of CTFT Lowpass Filter X(j) 1 H(j) Y(j) - 0 0 0, 0 Y( j) X ( j), 0 0 0, 0
Convolution Property of CTFT Highpass Filter: X(j) 1 H(j) Y(j) - 0 0 X( j), 0 Y( j) 0, 0 0 X( j), 0
Convolution Property of CTFT Bandpass Filter: X(j) 1 H(j) Y(j) - 2-1 1 2 X( j), 1 2 Y( j) X ( j), 1 2 0, otherwise
Examples
CTFT Table
CTFT Table
CTFT Table
Multiplication Property of CTFT 1 F r( t) s( t) p( t) R( j) S( j ) P( j( )) d 2 This is a dual of the convolution property
Multiplication Property of CTFT Idea of AM (amplitude modulation) FT of r(t) FT of p(t)=cos 0 t modulation g(t) = r(t)p(t) A R(j) P(j) G(j) A/2-0 0-0 0 demodulation g(t)p(t) F Q(j) Original signal is recovered after a low-pass filter A/2 A/4-2 0 0 2 0
Multiplication Property of CTFT A communication system A B N A B N modulation demodulation + bandpass filtering
Causal LTI Systems Described by Differential Equations
Linear Constant-Coefficient Differential Equations N 0 a d y( t) dt The DE describes the relation between the input x(t) and the output y(t) implicitly In this course, we are interested in DEs that describe causal LTI systems Therefore, we assume the initial rest condition 0 which also implies N 1 dy( t0) d y( t0) yt ( 0) 0 N 1 dt dt M b d x( t) dt If x( t) 0 for t t, then y( t) 0 for t t 0 0
Frequency Response What is the frequency response H(jw) of the following system? It is given by M N dt t x d b dt t y d a 0 0 ) ( ) ( 0 0 ( ) ( ) ( ) M N b j H j a j
Example Q) 2 d y t dy t dx t y t 2 dt dt dt ( ) ( ) ( ) 4 3 ( ) 2 x( t), t x( t) e u( t). A) Y ( j) H ( j) X ( j) j 2 1 2 ( j) 4( j) 3 j 1 j 2 2 ( j1) ( j3) 1 1 1 4 2 4 2 j 1 ( j 1) j 3 1 1 1 y t e te e u t 4 2 4 t t 3t ( ) ( )