2. CHAPTER 5 Fourier Transform
IR ROSDISHAM B. ENDUT
Dr. Soh Ping JACK
NOVEMBER 2017

3. analyze electric circuit
Course Outcome
CO2
Ability to
analyze electric circuit
using
Fourier Transform

4. Outline The concept of the Fourier transform
Fourier Transform Properties
The Laplace and Fourier transforms comparison
Derivation of Fourier transform for Non Periodic Functions
Circuit analysis using Fourier transform

5. Concept of Fourier Transform
The Fourier transform is an integral transform of a function, f(t) from the time domain to the frequency domain.
In general, F(ω) is a complex function.
Its magnitude is called the amplitude spectrum.
Its phase is called the phase spectrum.

6. Exponential Fourier Series
𝑒 +𝑗𝑛 𝜔 0 𝑡 =cos 𝑛 𝜔 0 𝑡+𝑗 𝑠𝑖𝑛 𝑛 𝜔 0 𝑡
𝑒 −𝑗𝑛 𝜔 0 𝑡 =cos 𝑛 𝜔 0 𝑡−𝑗 𝑠𝑖𝑛 𝑛 𝜔 0 𝑡

7. Example 1 (Ex.18.2 pg. 819)
Determine the Fourier transform of a rectangular pulse of wide t and height A, as shown below.

8. Solution If A = 10 and ᴦ = 2, then F(ω) = 20 sinc ω
* sinc(x) = cardinal sine function or sinc function

9. Example 2 (Ex pg. 819)
Obtain the Fourier transform of the exponential function as shown below.

10. Inverse Fourier Transform
Frequency domain to time domain:

11. Fourier Transform Properties - Linearity
Linearity: If F1(ω) and F2(ω) are the Fourier transforms of f1(t) and f2(t) then
Where a1 and a2 are constants.
This simply states that the transform of a linear combination of functions equals the linear combination of each transform.

12. Fourier Transform Properties - Time Scaling
In time scaling:
This shows that time expansion (|a|>1) corresponds to frequency compression and vice versa.
In other words, imagine a pulse that becomes shorter; the Fourier transform of this broadens in frequency.

13. Example - Time Scaling The effect of time scaling
Time compression results in:
Frequency Expansion and
Reduced Amplitude
Time Compression
Frequency Expansion

14. Fourier Transform Properties - Frequency Shifting
In frequency shifting:
ℱ 𝑓(𝑡) =𝐹(𝜔)
ℱ 𝑓(𝑡) 𝑒 𝑗 𝜔 0 𝑡 =𝐹(𝜔− 𝜔 0 )
A frequency shift in the frequency domain adds a phase shift to the time function.
This can also be seen as amplitude modulation in the time domain ( ).
Amplitude modulation (AM) is a process whereby the amplitude of the carrier is controlled by the modulating signal
This has important consequences for modulation, a common form of communication.

15. Fourier Transform Properties – Time Differentiation
In time differentiation:
The transform of the derivative of f (t) is obtained by multiplying the transform of f (t) by jω.
This can be generalized to the n’th derivative:

16. Fourier Transform Properties – Time Integration
In time integration:
the transform of the integral of f (t) is obtained by dividing the transform of f (t) by jω and adding the result to the impulse term that reflects the dc component F(0).
Note that the upper bound of the integral is t.
If it were not, the Fourier transform would be of a constant.

17. Fourier Transform Properties – Reversal
In reversal:
This property states that reversing f (t) about the time axis reverses F(ω) about the frequency axis.
This can be considered a special case of time scaling if a = -1.

18. Fourier Transform Properties - Duality and Convolution
Duality states
This expresses the symmetry property of the Fourier transform.
Convolution states:
Operation on two functions (h and x); it produces a third function,

19. Fourier Transform Properties - Convolution

20. Fourier Transform Pairs

21. More Fourier Transform Pairs

22. Relationship between Laplace and Fourier Transform
The Laplace transform is one-sided in that the integral is over 0 < t <  making it only useful for positive time functions, f (t), t > 0.
The Fourier transform is applicable to functions defined for all time.
For a function that is nonzero for positive time only, the two transforms are related by:
Therefore, this equation shows that the Laplace transform is related to the entire s plane, whereas the Fourier transform is restricted to the jω axis.

23. The Laplace transform is applicable to a wider range of functions than the Fourier transform.
The Laplace transform is better suited for the analysis of transient problems involving initial conditions, since it permits the inclusion of the initial conditions, whereas the Fourier transform does not.

24. Aperiodic Function of Fourier Transform
Fourier series can represent any periodic waveform.
But, many signals of interest in electronics are not periodic.
Although these cannot be represented in a Fourier series, they can be transformed into frequency domain by use of something called the Fourier transform.

25. Stretching the Period
One way to consider a non-periodic function is to take a periodic one and stretch the period.
Consider the periodic function shown at the bottom of the figure.
If the period T→, then the function becomes non-periodic.

27. Effect on the Spectrum The impact of increasing the period,
the pulses are spaced out more,
the peaks in the frequency spectrum get closer together
The amplitude also drops.

28. Circuit Applications
Apply Fourier transforms to circuits with Ohm’s law
The expressions for impedances as in phasor analysis
Fourier transforms can’t handle initial conditions.
The transfer function again defined as.

29. Circuit Analysis using Fourier Transform
Fourier transforms can be applied to circuits with non-sinusoidal excitation in exactly the same way as phasor techniques being applied to circuits with sinusoidal excitations.
By transforming the functions for the circuit elements into the frequency domain and take the Fourier transforms of the excitations, conventional circuit analysis techniques could be applied to determine unknown response in frequency domain.
Finally, apply the inverse Fourier transform to obtain the response in the time domain.

30. Example (Ex pg.834)
Calculate v0(t) using Fourier transform for the circuit shown below if the input voltage is vi(t)=2e-3tu(t)

31. The Fourier transform of the input voltage,
(refer Table 18.2 pg.829)
The transfer function of the given circuit by VD:
Hence
Applying PFE and the inverse Fourier transform

33. Example
Calculate i0(t) using Fourier transform for the circuit shown below if the input voltage is is(t) = e-tu(t)

34. Parseval’s Theorem
Parseval’s theorem relates the energy (W) carried by a signal to the Fourier transform of the signal.
If p(t) is the power associated with the signal, the energy carried is:
For convenience, we can make a comparison to the energy content of a voltage or current passing through a 1Ω load.

35. The energy delivered to the resistor is:
Parseval’s theorem states that the total energy delivered to a 1Ω resistor equals the total area under the square of f (t) or 1/2 times the total area under the square of the magnitude of the Fourier transform of f(t).

36. Alternatively, one can integrate from zero to infinity since |F(ω)|2 is an even function:
It is also possible to calculate the energy in any frequency band:

37. Example
The current through a 1 Ω resistor is 10e-2|t| u(t) A. Calculate the total energy absorbed in the resistor.

38. End of Chapter 5