Fourier Series and Transform KEEE343 Communication Theory Lecture #7, March 24, 20 Prof. Young-Chai Ko koyc@korea.ac.kr
Summary Fourier transform Properties Fourier transform of special function Fourier transform of periodic signals
Properties of Fourier Transform Linearity Dilation Conjugation rule Duality property Time shifting property Frequency shifting property Differentiation in the time domain Modulation theorem Convolution theorem Correlation theorem Rayleigh s Energy theorem (or Parserval s theorem) Area property
Time shifting property If g(t)! G(f), then g(t t 0 )! G(f)exp( j2 ft 0 ) Z F[g(t t 0 )] = g(t t 0 )exp( j2 ft) dt Change of variable: t t 0 = F[g(t t 0 )] = Z = exp( j2 ft 0 ) g( )exp[ j2 f( + t 0 )] d Z = G(f)exp( j2 ft 0 ) g( )exp( j2 f ) d Time shift does not change the amplitude spectrum but the phase is changed by 2 ft 0
Frequency shifting property If g(t)! G(f), then exp(j2 f c t)g(t)! G(f f c ) F [exp(j2 f c )g(t)] = = Z Z g(t)exp(j2 f c t)exp( g(t)exp( j2 (f f c )t) dt j2 ft) dt Hence, exp(j2 f c t)g(t)! G(f f c ) We can also show that exp( j2 f c t)g(t)! G(f + f c )
Example of Frequency Shifting Property Find the FT of radio frequency pulse given as t g(t) =rect cos(2 f c t) T Using the Euler s formula we have cos(2 f c t)= 2 [exp(2 f ct)+exp( j2 f c t)] Then using the frequency shifting property of the Fourier transform we get the desired result: rect t T cos(2 f c t) T 2 {sinc[t (f f c)] + sinc[t (f + f c )]}
[Ref: Haykin & Moher, Textbook]
Area property Z g(0) = g(t) dt = G(0) Z G(f) df Differentiation in the time domain If g(t)! G(f), then d g(t)! j2 fg(f) dt and d n dt n g(t)! (j2 f)n G(f)
Modulation theorem Let g (t)! G (f), and g 2 (t)! G 2 (f), then g (t)g 2 (t)! Z G ( )G 2 (f ) d Convolution Theorem Z g ( )g 2 (t ) d! G (f)g 2 (f) g (t) g 2 (t)! G (f)g 2 (f)
In the special case of f c T>>, that is, the frequency f c 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 f c)], f > 0 0, f =0, T 2 sinc[t (f + f c)], f < 0
Correlation theorem Z g ( )g 2(t ) d! G (f)g 2(f) Rayleigh s Energy theorem Z g(t) 2 dt = Z G(f) 2 df
[Ref: Haykin & Moher, Textbook]
Inverse Relationship Between Time and Frequency If the time-domain description of a signal is changed, the frequency-domain description of the signal is changed in an inverse manner, and vice versa. If a signal is strictly limited in frequency, the time-domain description of the signal will trail on indefinitely, even though its amplitude may assume a progressively smaller value - a signal cannot be strictly limited in both time and frequency.
Bandwidth A measure of extent of the significant spectral content of the signal for positive frequencies. Commonly used three definitions. Null-to-null bandwidth 2. 3-dB bandwidth 3. Root mean-square (rms) bandwidth W rms = R f 2 G(f) 2 df R G(f) 2 df! 2
Time-Bandwidth Product The product of the signal s duration and its bandwidth is always a constant (duration) (bandwidth) = constant Define the time rms duration R T rms = t2 g(t) 2 dt R g(t) 2 dt! /2 Then we can show the time-bandwidth product with the following form T rms W rms 4
Dirichlet conditions Fourier transform is only applicable to time functions that satisfy the Dirichlet conditions such that Z g(t) 2 dt < However, it would be highly desirable to extend the theory in two ways:. To combine the theory of the Fourier series and transform into a unified framework, so that the Fourier series may be treated as a special case of the Fourier transform. 2. To expand the applicability of the Fourier transform to include power signals - that is, the signals for which the condition holds: lim T! 2T Z T T g(t) 2 dt <
Dirac Delta Function Dirac delta function having zero amplitude everywhere except at t=0, where it infinitely large in such a way that it contains unit area under its curve. Integral of the product Z (t) =0, t 6= 0 Z (t) dt = g(t) (t t 0 ) g(t) (t t 0 ) dt = g(t 0 ) Convolution Z g( ) (t ) dt = g( ) =) g(t) (t) =g(t)
Fourier transform of the delta function F [ (t)] = Z Delta function as a limiting form of the Gaussian pulse g(t) = exp t 2 2 (t)exp( j2 ft) dt = 4 3.5 =0.25 3 2.5 g(t) 2.5 =0.5 0.5 = =2 0 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 t
Fourier transform of Gaussian pulse G(f) =exp( 2 f 2 ) 0.9 =0.25 0.8 0.7 =0.5 0.6 G(f) 0.5 0.4 = 0.3 0.2 =2 0. 0 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 f
Application to the Delta Function DC signal Using the duality property we can show that! (f) which gives the following relation Z exp( j2 ft) dt = (f) Recognizing that the delta function is real valued, we can simplify this relation as Z cos(2 f t) dt = (f)
Complex exponential function exp(j2 f c t)! (f f c ) Sinusoidal functions cos(2 f c t)= 2 [exp(j2 f ct)+exp( j2 f c t)] cos(2 f c t)! 2 [ (f f c)+ (f + f c )] Similarly, sin(2 f c t)! 2j [ (f f c) (f + f c )]