Fourier Series and Transform KEEE343 Communication Theory Lecture #6, March 22, 2011 Prof.Young-Chai Ko koyc@korea.ac.kr Summary ·•Fourier transform ·• Definition ·• Amplitude and Phase Spectra ·• Properties ·• Examples Fourier Transform x(t) Suppose that is nonperiodic but is an energy signal, so that it ·•is integrable square in the interval . ( 1, 1) 1 ·• x(t) |t| T0 In the interval , we can represent as the Fourier 2 series 1 X x(t) = n= 1 " 1 T0 Z T0 /2 x( )e j2⇥nf0 d T0 /2 # ej2⇥nf0 t f = 1/T ·•where . x(t) T !1 ·•To represent for all time, we simply let such that 0 0 0 nf0 = n/T0 1/T0 P summation ! ! ! continuous variable, f the differential, df R integral • Thus x(t) = • Z 1 1 Z 1 x( )e j2⇥f d ej2⇥f t df 1 Defining the inside integral as Z 1 X(f ) = x( )e j2⇥f d 1 • we can write x(t) = Z 1 X(f )ej2 ft df 1 x(t) Fourier Transform X(f ) X(f ) Inverse Fourier Transform x(t) • Notations X(f ) = F [x(t)] x(t) = F 1 [X(f )] x(t) () X(f ) Amplitude and Phase Spectra • • • Writing X(f ) in phasor form: X(f ) = |X(f )|ej , (f ) = \X(f ) we can show that for real x(t) , that |X(f )| = |X( f )| and ( f ) = (f ) This is done by Euler’s theorem to write R I • (f ) = = <X(f ) = =X(f ) = Z 1 x(t) cos(2 f t) dt 1 1 Z x(t) sin(2 f t) dt 1 Then, the square of amplitude and the phase are |X(f )|2 = R2 + I 2 , ✓ ◆ I (f ) = tan 1 R • Amplitude spectrum: Plot of |X(f )| versus f • Phase spectrum: Plot of \X(f ) versus f Example • ✓ ◆ t Fourier transform of rectangular pulse g(t) = A rect T Z 1 Z T /2 G(f ) = g(t) exp( j2 f t) dt = A A exp( j2 f t) dt 1 = = = = = = 1 A exp( j2 f t) j2 f T /2 t=T /2 t= T /2 A [ exp( j2 f T /2) exp(j2 f T /2)] j2 f ✓ ◆ A exp(j f T ) exp( j f T ) f 2j ✓ ◆ sin( f T ) A f ✓ ◆ sin( f T ) AT fT AT sinc( f T ) Sinc Function • Definition sinc(x) = sin( x) x [Ref: Haykin & Moher, Textbook] Fourier Transform of Rectangular Pulse • Rectangular pulse with the width of T and the height of A so that the area is at the center of zero Y✓ t ◆ A T ! AT sinc(f T ) [Ref: Haykin & Moher, Textbook] Fourier Transform of Exponential Function • Exponential function such as g(t) = exp( • t)u(t) Fourier transform G(f ) = F [g(t)] Z 1 = g(t)e j2⇥f t dt = 1 1 + j2⇥f Similarly for g(t) = exp(+ t)u( t) = • G(f ) = 1 j2⇥f Z 1 0 e ( +j2⇥f )t dt Properties of the Fourier Transform • Linearity (Superposition) property then for all constants c1 and c2 c1 g1 (t) + c2 g2 (t) • Dilation property g(at) • Proof: F [g(at)] ! c1 G1 (f ) + c2 G2 (f ) ✓ ◆ 1 f ⇥ G |a| a Z 1 = g(at) exp( j2 f t) dt 1 = = change of variable:⇥ = at ✓ ◆ Z 1 1 f g(⇥ ) exp j2 ⇥ a 1 a ✓ ◆ 1 f G a a d⇥ • Reflection property g( t) ! G( f ) [Ref: Haykin & Moher, Textbook] • Conjugation rule Let g(t) ! G(f ) , then for a complex-valued time function g(t) g ⇤ (t) ! G⇤ ( f ) Prove this: g ⇤ ( t) ! G⇤ (f ) • Duality property If g(t) ! G(f ) , then G(t) ! g( f ) Z 1 g( t) = G(f ) exp( j2 f t) df 1 which can be expanded part in going from the time domain to the frequency domain: Example of Dual Property: Sinc Pulse • We have the following pair of the Fourier transform: g(t) = A sinc(2W t) • A ! G(f ) = rect 2W ✓ f 2W ◆ Then, if the time function, given as h(t) = G(t) = A rect 2W ✓ t 2W ◆ ! H(f ) = g( f ) = A sinc( 2W f ) = A sinc(2W f ) • Time shifting property If g(t) • t0 ) ! G(f ) exp( j2 f t0 ) Frequency shifting property If g(t) • ! G(f ) , then g(t ! G(f ) , then exp(j2 fc t)g(t) Area property under g(t) If g(t) ! G(f ) , then Z 1 1 g(t) dt = G(0) ! G(f fc ) Example of Frequency Shifting Property • Find the FT of radio frequency pulse given as ✓ ◆ t g(t) = rect cos(2 fc t) T Using the Euler’s formula we have 1 cos(2 fc t) = [exp(2 fc t) + exp( j2 fc t)] 2 Then using the frequency shifting property of the Fourier transform we get the desired result: ✓ ◆ t T rect cos(2 fc t) ⇥⇤ {sinc[T (f T 2 fc )] + sinc[T (f + fc )]} [Ref: Haykin & Moher, Textbook] • In the special case of fc T >> 1, that is, the frequency fc is large compared to the reciprocal of the pulse duration T - we may use the approximate result G(f ) = 8 < : T 2 sinc[T (f T 2 sinc[T (f + fc )], fc )], 0, f >0 f = 0, f <0 • • Area property under G(f ) Z g(0) = 1 G(f ) df 1 Differentiation in the time domain If g(t) ! G(f ) , then d g(t) dt ! j2 f G(f ) and dn g(t) dtn ! (j2 f )n G(f ) • Modulation theorem Let g1 (t) ! G1 (f ), and g2 (t) g1 (t)g2 (t) • ! Z 1 ! G2 (f ) , then G1 ( )G2 (f )d 1 Convolution Theorem Z 1 g1 ( )g2 (t 1 g1 (t) ⇤ g2 (t) )d ! G1 (f )G2 (f ) ! G1 (f )G2 (f ) • Correlation theorem Z • 1 1 g1 ( )g2⇤ (t )d ! G1 (f )G⇤2 (f ) Rayleigh’s Energy theorem Z 1 1 |g(t)|2 dt = Z 1 1 |G(f )|2 df [Ref: Haykin & Moher, Textbook]