Continuous-Time Fourier Transform

Transcription

1 Signals and Systems Continuous-Time Fourier Transform Chang-Su Kim continuous time discrete time periodic (series) CTFS DTFS aperiodic (transform) CTFT DTFT

2 Lowpass Filtering Blurring or Smoothing Original average Strong LPF e.g. 21x21 moving average filter Less strong LPF e.g. 11x11 moving average filter

3 Highpass Filtering Edge Extraction

4 CTFT Formula and Its Derivation

5 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

6 Bridge Between Fourier Series and Transform Setch a on the -axis a 2T 1 /T 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).

7 Bridge Between Fourier Series and Transform The same setch a on the -axis: a 2T 1 /T On the -axis, the distance between two consecutive a s is 0 =2/T, which is the fundamental frequency.

8 Bridge Between Fourier Series and Transform The same setch Ta on the -axis: Ta 2T 1 X(jw) 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

9 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

10 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

11 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

12 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)

13 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

14 Some Examples Ex 1) Impulse function constant function 1 0 t x( t) ( t) F X ( j) ( t) e jt dt 1

15 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

16 More Examples

17 Unified Framewor for CTFS and CTFT: Periodic Signals Can Also Be Represented as Fourier Transform

18 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 a 3

19 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 ( j) ( j) /j /j

20 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

21 Properties of CTFT

22 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

23 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

24 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

25 Convolution Property of CTFT Lowpass Filter X(j) 1 H(j) Y(j) , 0 Y( j) X ( j), 0 0 0, 0

26 Convolution Property of CTFT Highpass Filter: X(j) 1 H(j) Y(j) X( j), 0 Y( j) 0, 0 0 X( j), 0

27 Convolution Property of CTFT Bandpass Filter: X(j) 1 H(j) Y(j) X( j), 1 2 Y( j) X ( j), 1 2 0, otherwise

28 Examples

29 CTFT Table

30 CTFT Table

31 CTFT Table

32 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

33 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/ demodulation g(t)p(t) F Q(j) Original signal is recovered after a low-pass filter A/2 A/

34 Multiplication Property of CTFT A communication system A B N A B N modulation demodulation + bandpass filtering

35 Causal LTI Systems Described by Differential Equations

36 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

37 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

38 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 ( j) 4( j) 3 j 1 j 2 2 ( j1) ( j3) j 1 ( j 1) j y t e te e u t t t 3t ( ) ( )