signals and systems, introduction.pptx

signals and systems, introduction.pptx

1. Fourier Analysis of Signals and Systems Dr. Babul Islam Dept. of Applied Physics and Electronic Engineering University of Rajshahi 1

2. Outline Response of LTI system in time domain Properties of LTI systems Fourier analysis of signals Frequency response of LTI system 2

3. A system satisfying both the linearity and the time-invariance properties. LTI systems are mathematically easy to analyze and characterize, and consequently, easy to design. Highly useful signal processing algorithms have been developed utilizing this class of systems over the last several decades. They possess superposition theorem. Linear Time-Invariant (LTI) Systems 3

4. Linear System: + T + T T System, T is linear if and only if i.e., T satisfies the superposition principle. 4

5. Time-Invariant System: A system T is time invariant if and only if T implies that T Example: (a) Since , the system is time-invariant. (b) Since , the system is time-variant. 5

6. Any input signal x ( n ) can be represented as follows: Consider an LTI system T . 1 n 1 2 -1 -2 … … Graphical representation of unit impulse. T T Now, the response of T to the unit impulse is T Applying linearity properties, we have 6

7. LTI system can be completely characterized by it’s impulse response. Knowing the impulse response one can compute the output of the system for any arbitrary input. Output of an LTI system in time domain is convolution of impulse response and input signal, i.e., T (LTI) Applying the time-invariant property, we have 7

8. Properties of LTI systems (Properties of convolution) Convolution is commutative x[n]  h[n] = h[n]  x[n] Convolution is distributive x[n]  (h 1 [n] + h 2 [n]) = x[n]  h 1 [n] + x[n]  h 2 [n] 8

9. Convolution is Associative: y[n] = h 1 [n]  [ h 2 [n]  x [n] ] = [ h 1 [n]  h 2 [n] ]  x [n] h 2 x[n] y[n] h 1 h 2 x[n] y[n] h 1 = 9

10. Frequency Analysis of Signals Fourier Series Fourier Transform Decomposition of signals in terms of sinusoidal or complex exponential components. With such a decomposition a signal is said to be represented in the frequency domain. For the class of periodic signals, such a decomposition is called a Fourier series. For the class of finite energy signals ( aperiodic ), the decomposition is called the Fourier transform. 10

11. Consider a continuous-time sinusoidal signal, This signal is completely characterized by three parameters: A = Amplitude of the sinusoid  = Angular frequency in radians/sec = 2 f  = Phase in radians Fourier Series for Continuous-Time Periodic Signals: A A cos   t 11

12. Complex representation of sinusoidal signals: Fourier series of any periodic signal is given by: Fourier series of any periodic signal can also be expressed as: where where 12

13. Example: 13

14. Power Density Spectrum of Continuous-Time Periodic Signal: This is Parseval’s relation. represents the power in the n -th harmonic component of the signal. Power spectrum of a CT periodic signal. If is real valued, then , i.e., Hence, the power spectrum is a symmetric function of frequency. 14

15. Define as a periodic extension of x ( t ): Fourier Transform for Continuous-Time Aperiodic Signal: Assume x ( t ) has a finite duration. Therefore, the Fourier series for : where Since for and outside this interval, then 15

16. Now, defining the envelope of as Therefore, can be expressed as As Therefore, we get 16

17. Energy Density Spectrum of Continuous-Time Aperiodic Signal: This is Parseval’s relation which agrees the principle of conservation of energy in time and frequency domains. represents the distribution of energy in the signal as a function of frequency, i.e., the energy density spectrum. 17

18. Fourier Series for Discrete-Time Periodic Signals: Consider a discrete-time periodic signal with period N . Now, the Fourier series representation for this signal is given by where Since Thus the spectrum of is also periodic with period N . Consequently, any N consecutive samples of the signal or its spectrum provide a complete description of the signal in the time or frequency domains. 18

19. Power Density Spectrum of Discrete-Time Periodic Signal: 19

20. Fourier Transform for Discrete-Time Aperiodic Signals: The Fourier transform of a discrete-time aperiodic signal is given by Two basic differences between the Fourier transforms of a DT and CT aperiodic signals. First, for a CT signal, the spectrum has a frequency range of In contrast, the frequency range for a DT signal is unique over the range since 20

21. Second, since the signal is discrete in time, the Fourier transform involves a summation of terms instead of an integral as in the case of CT signals. Now can be expressed in terms of as follows: 21

22. Energy Density Spectrum of Discrete-Time Aperiodic Signal: represents the distribution of energy in the signal as a function of frequency, i.e., the energy density spectrum. If is real, then (even symmetry) Therefore, the frequency range of a real DT signal can be limited further to the range 22

23. 23 Frequency Response of an LTI System For continuous-time LTI system For discrete-time LTI system

24. Conclusion The response of LTI systems in time domain has been examined. The properties of convolution has been studied. The response of LTI systems in frequency domain has been analyzed. Frequency analysis of signals has been introduced. 24