The Continuous Time Fourier Transform

Transcription

1 COMM 401: Signals & Systems Theory Lecture 8 The Continuous Time Fourier Transform

2 Fourier Transform Continuous time CT signals Discrete time DT signals Aperiodic signals nonperiodic periodic signals Aperiodic signals nonperiodic periodic signals

3 Fourier Representation of Signals

4 Development of FT of an aperiodic signal

5 Continuous Time Fourier Transform The previous equation can be seen as samples of an envelope function as: Let k0 2sin T / represents the envelope of coefficients Envelope With thought of as a continuous variable, the function 1 a k Ta and the are simply equally spacedsamples of this envelope k

6 Continuous Time Fourier Transform ~ the set of F.S Coefficients approaches the Envelope function

7 Derivation of Continuous Time Fourier Transform Envelope

8 Fourier Transform & Inverse FT use T 2 0

9 T 2 0

10 Conditions of Convergence of FT Convergence is guaranteed if xt satisfies the following conditions for non-periodic signals:

11 Example: Analog Signals Fourier Transform

12 Remember: The sinc function

13 Example : Determine the FT of the signal? Famous Signal

14 Example: Solution As W increases the width main lobe of x t becomes narrower and the main peak becomes higher. of

15 Example: Determine the FT of the signal? a > 0 Famous Signal Draw the magnitude and phase of FT of the signal.

16 Example: Magnitude Response Phase Response

17 Example: Determine the FT of the signal? Famous Signal 1

18 Example: Determine the IFT of the signal?

19 Properties of the Continuous Time FT Remarks

20 Example: Determine the FT of the signal? 1

21 Properties of the Continuous Time FT Time Scale

22 Properties of the Continuous Time FT xt = x ev t + x od t and hence, Example: x t X j Ev Od x t ReX j x t j ImX j Famous Signal

23

24 Properties of the Continuous Time FT

25 Properties of the Continuous Time FT

26 Example:

27 Properties of the Continuous Time FT

28 Properties of the Continuous Time FT xt X j LTIS ht H j yt Y j Convolution Property: yt ht*xt F.T Y j H j X j Example : Consider a LTIS with impulse responseh t : Then the output of the system in the time domain is : y t x t* h t and in the frequency domain : Y j H j X j Important

29 Example: , 1 1 Solution input the to response: impulse system with a of output the Find t u e e t u e t u e t y j j j B j A j j j Y j j X j j H j X j H j Y t u e t x t u e t h t t t t t t Important

30 Properties of the Continuous Time FT Duality: By comparing the F.T and the inverse F.T equations, we observe that these equations are similar but not identical in form. This symmetry leads to a property of the F.T referred to as duality. Example:

31 Specifically, because of the symmetry between F.T and IFT Equations for any transform pair, there is a dual pair with the time and frequency variables interchanged. We saw that differentiation in the time domain corresponds to multiplication by in the frequency domain. Then, multiplication by -jt in the time domain corresponds roughly to differentiation in the frequency domain. If we differentiate the Fourier transform equation w.r.t : X dx j d Then; j x t e jtx t jt dt jtx t e FT jt dt dx j d

32 Properties of the Continuous Time FT Similar to Time shifting Property Similar to Integration Property j X e t t x o t j FT o 0 1 X j X j d x t Similar to differentiation Property Frequency domain Multiplication Property: ] [ 2 1 j P j S j R t p t s t r

33 Example: t zt=xt j a j 2

34 Frequency Shaping and Filtering Frequency Response: e j

35 Frequency Shaping and Filtering

36 Frequency Shaping and Filtering

37 Frequency Response and Filtering

38 Frequency Response and Filtering

39 Practical Example: Remember Frequency Response and Filtering a 1 RC

40 Frequency Response and Filtering

41 Properties of the Continuous Time FT Convolution property: h1 t h2 t H1 j H 2 j

42 Frequency Response and Filtering

43 Example: The frequency response of this system is the Fourier transform of its impulse response which is given by: Then, for any input xt, the Fourier Transform of the output is given by: In time domain and using time shifting property, yt is given by:

44 Example: