COMM 401: Signals & Systems Theory Lecture 8 The Continuous Time Fourier Transform
Fourier Transform Continuous time CT signals Discrete time DT signals Aperiodic signals nonperiodic periodic signals Aperiodic signals nonperiodic periodic signals
Fourier Representation of Signals
Development of FT of an aperiodic signal
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
Continuous Time Fourier Transform ~ the set of F.S Coefficients approaches the Envelope function
Derivation of Continuous Time Fourier Transform Envelope
Fourier Transform & Inverse FT use T 2 0
T 2 0
Conditions of Convergence of FT Convergence is guaranteed if xt satisfies the following conditions for non-periodic signals:
Example: Analog Signals Fourier Transform
Remember: The sinc function
Example : Determine the FT of the signal? Famous Signal
Example: Solution As W increases the width main lobe of x t becomes narrower and the main peak becomes higher. of
Example: Determine the FT of the signal? a > 0 Famous Signal Draw the magnitude and phase of FT of the signal.
Example: Magnitude Response Phase Response
Example: Determine the FT of the signal? Famous Signal 1
Example: Determine the IFT of the signal?
Properties of the Continuous Time FT Remarks
Example: Determine the FT of the signal? 1
Properties of the Continuous Time FT Time Scale
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
Properties of the Continuous Time FT
Properties of the Continuous Time FT
Example:
Properties of the Continuous Time FT
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
Example: 2 1 1 1 2 1 2 1 1 2 1, 1 1 Solution input the to response: impulse system with a of output the Find 2 2 2 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
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:
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
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
Example: t zt=xt j a j 2
Frequency Shaping and Filtering Frequency Response: e j
Frequency Shaping and Filtering
Frequency Shaping and Filtering
Frequency Response and Filtering
Frequency Response and Filtering
Practical Example: Remember Frequency Response and Filtering a 1 RC
Frequency Response and Filtering
Properties of the Continuous Time FT Convolution property: h1 t h2 t H1 j H 2 j
Frequency Response and Filtering
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:
Example: