Fourier Series and Transform KEEE343 Communication Theory Lecture #7, March 24, Prof. Young-Chai Ko

Transcription

1 Fourier Series and Transform KEEE343 Communication Theory Lecture #7, March 24, 20 Prof. Young-Chai Ko

2 Summary Fourier transform Properties Fourier transform of special function Fourier transform of periodic signals

3 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

4 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

5 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 )

6 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 )]}

7 [Ref: Haykin & Moher, Textbook]

8 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)

9 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)

10 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

11 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

12 [Ref: Haykin & Moher, Textbook]

13 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.

14 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

15 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

16 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 <

17 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)

18 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 = = g(t) 2.5 = = = t

19 Fourier transform of Gaussian pulse G(f) =exp( 2 f 2 ) 0.9 = = G(f) = = f

20 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)

21 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 )]

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