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The Fourier Transform and Its Applications

ENT 272: Signal Theory and Applications Sem. I (10/11). The Fourier Transform and Its Applications. System Analysis with the Fourier Transform. LTI System, h(t) or H( ω ). H( ω ) is called the frequency response/ Transfer function of the system.

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The Fourier Transform and Its Applications

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  1. ENT 272: Signal Theory and Applications Sem. I (10/11) The Fourier Transform and Its Applications

  2. System Analysis with the Fourier Transform LTI System, h(t) or H(ω) H(ω) is called the frequency response/ Transfer function of the system

  3. System Analysis with the Fourier Transform We can write the Transfer Function as where h(t) is the impulse response of the system. H(ω) is the Fourier Transform of h(t). H(ω) can berepresented in polar form as where A(ω) is called the amplitude response and (ω) is called the phase response

  4. System Analysis with the Fourier Transform Previously, we have a system that can be represented in terms of its impulse response or its transfer function as LTI System, h(t) or H(ω) where h(t) : Impulse reponse H(ω) : Frequency response or Transfer function and

  5. System Analysis with the Fourier Transform Since H(ω) is, in general, a complex quantity, we can write it as Where If H(ω) is the F.T of a real time function h(t) , it follows that

  6. System Analysis with the Fourier Transform Fourier Transform can be used to simplify the calculation of the response linear system to input signal. For example, we use Fourier Transform to analyze the system that are describe by linear, time invariant differential equations.

  7. System Analysis with the Fourier Transform Example 1: Consider the circuit shown in Figure below, where Vin is the input signal and VL is the output signal of the circuit.

  8. System Analysis with the Fourier Transform This circuit cam be simplified by the differential equations as follows: Then we take the Fourier Transform of each equation by using Fourier Transform properties, we get:

  9. System Analysis with the Fourier Transform Then solve for I(ω): Substitute I(ω) into the Vout equation we get: We call H(ω) as a transfer function/ frequency response of the system.

  10. System Analysis with the Fourier Transform So we can define a function: The input-output relationship of the system is: When we compare the above equation with the block diagram in the first slide, we see the equations is relates the output voltage output of the signal to the input voltage of the system

  11. Application of Fourier Transform • Consider below concept: Where Vin() is the Fourier Transform of the input signal to a system and Vout() is Fourier Transform of the output Signal. • The concept can be applied to design a filter which is always use in electric Signal Processing application. • Four type of ideal filter: • Ideal low-pass filter • Ideal high-pass filter • Ideal band-pass filter • Ideal band-stop filter

  12. Application of Fourier Transform • What is filter? Filter are used to elimanate unwanted components of Signal. Example show in Figure in next slide.

  13. Application of Fourier Transform • Real Filter Figure in next slide shows the schematic diagram of an RC low-pass filter. This filter is approximate same as the ideal low-pass filter. To show it we can anlyze the cirsuit by finding the frequency response of this circuit.

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