بسم الله الرحمن الرحيم University of Khartoum Department of Electrical and Electronic Engineering Third Year – 2015 Dr. Iman AbuelMaaly Abdelrahman

The Fourier Transform Bridge to Fourier Transform Convergence of Fourier Transform Exercises

3 Bridge to Fourier Transform Periodic Signals  Fourier Series Aperiodic Signals  Fourier Transform 2015

Bridge to Fourier Transform 2015 Periodic Signal Aperiodic Signal Fourier Series Fourier Transform T increases T  ∞ T T= 4 T 1 T= 16T 1 T ∞ As T increases decreases

5 Fourier Series 2015 Fourier Transform For Periodic Signals For aperiodic Signals Prove it

6 Time Domain vs. Frequency Domain Fourier Analysis (Series or Transform) is, in fact, a way of determining a given signal’s frequency content, i.e. move from time-domain to frequency domain. 2015

7 The Fourier Transform Given a signal x (t) in time-domain, its Fourier Transform X(j  ) is called as its “frequency spectrum”. The Fourier Transform of x(t) is: F { x(t) } = X (jω) 2015

8 Inverse Fourier Transform It is always possible to move back from the frequency-domain to time-domain, by Inverse Fourier Transform. The Inverse Fourier Transform of X (jω) is: F -1 { X (jω) } = x(t) 2015

9 Time and frequency domains x(t) X (jω) F Signal in time domain مجال الزمن Signal in frequency domain مجال التردد ( الطيف) 2015

10 Fourier Transform Pair of Equations The Fourier Transform of x(t): F { x (t)} Inverse Fourier Transform of X (jω): F -1 { X (jω)} 2015

11 Complex Spectrum If X(j  ) is complex, then the frequency spectrum is observed by: its magnitude | X(j  ) | and phase  X(j  ) plots 2015

12 Convergence of the Fourier Transform There are two important classes of signals for which the Fourier transform converges. 1.Signals of finite total energy, i.e. 2.Signals that satisfies the Dirichlet Conditions 2015

13 Dirichlet Conditions 1. The signal must be absolutely integrable 2. Over a finite interval of time, the signal must have finite number of maxima and minima (or variations) 3. Over a finite interval of time, the signal must have finite number of discontinuities. Also, those discontinuities must be finite. 2015

14 Example1 Compute the Fourier Transform of the following signal: a is complex in general. 2015

Solution: Use the Fourier Transform Equation ∞ 15

16 | X(j  ) | and  X(j  )Here’s a plot of the magnitude and phase of X(jω), | X(j  ) | and  X(j  ) For the important case a >0 real. 2015

17 The Inverse Fourier Transform Example 3 Calculate the inverse Fourier Transform of the following signal: 2015

18 Solution 2015

Solution

Fourier Transform Properties 1. Linearity 2. Time-Shifting 3. Frequency-Shifting 4.Time/Frequency Scaling 5. Time-Flip 6. Differentiation in Time

Fourier Transform Properties 7. Integration in Time 8. Differentiation in Frequency 9. Conjugate and symmetry 10. Convolution 11.The Multiplication property 12. Duality Property

22 Fourier Transform Properties 1. Linearity: Given two signals and Where a and b are any coefficients 2015

23 2. Time-Shifting Given A time shift results in a phase shift in the Fourier transform Fourier Transform Properties 2015

24 Fourier Transform Properties 3. Frequency-Shifting: Given 2015

Fourier Transform Properties 4.Time/Frequency Scaling Given 2015 Scaling the time variable either expands or contracts the Fourier Transform 25

For α >1, the signal x(αt) is sped up ( or compressed in time) Fourier transform of the signal expands to higher frequencies. When the signal is slowed down (α <1), the Fourier transform gets compressed to lower frequencies. Fourier Transform Properties

27 Fourier Transform Properties 5. Time-Flip Given then 2015

28 Fourier Transform Properties 6. Differentiation in Time: Given 2015 Differentiating a signal results in a multiplication of the Fourier transform by jω

29 Fourier Transform Properties 7. Integration in Time 2015 Integrating a signal results in a division of the Fourier transform by jω Prove this property

30 Fourier Transform Properties 8. Differentiation in Frequency: Given then 2015

31 9. Conjugate and symmetry Given In general if the signal is complex, its conjugate is x * (t), so we can take its conjugate and we obtain Fourier Transform Properties 2015

32 If In particular if the signal is real, i.e., then the Fourier transform has conjugate symmetry Fourier Transform Properties 2015

33 Fourier Transform Properties 2015

34 Fourier Transform Properties 10. Convolution Convolution in time-domain corresponds to multiplication in frequency-domain: 2015

35 Fourier Transform Properties 11.The Multiplication property Similarly, multiplication in time-domain corresponds to convolution in frequency-domain: 2015

Fourier Transform Properties The total energy in an aperiodic signal is equal to the total energy in its spectrum. 11. Parseval’s Theorem

Fourier Transform Properties 11. Parseval’s Theorem Total Energy in a signal Is the energy per unit time Is the energy per unit frequency (1) 37 Prove this property 2015

The Energy Density Spectrum is the energy-density spectrum We can find the energy of a signal in a given frequency band by integrating its energy- density spectrum in the interval of frequencies. (2)

12. Duality Property The Fourier transform pair is quite symmetric. This results in a duality between the time domain and the frequency domain Fourier Transform Properties

12. Duality Property Example

Bode Plot A Bode plot is a graph of the transfer function of a linear, time-invariant system versus frequency, plotted with a log-frequency axis, to show the system's frequency response. It is usually a combination of a Bode magnitude plot, expressing the magnitude of the frequency response gain, and a Bode phase plot, expressing the frequency response phase shift.

Basic Continuous-Time Fourier Transform Pairs

Basic Continuous-Time Fourier Transform Pairs

Next Lecture Applications of Fourier Transform