Lecture 7: Discrete-Time Fourier Transform

1. Lecture 7:Discrete-Time Fourier Transform Instructor: Dr. Ghazi Al Sukkar Dept. of Electrical Engineering The University of Jordan Email: ghazi.alsukkar@ju.edu.jo

2. Outline Frequency domain Representation of Discrete Time Signals and Systems. Frequency response of an LTI system Discrete-time Fourier Transform.

3. Frequency Response of LTI Systems • Complex exponentials are eigenfunctions of LTI systems: • Let’s see what happens if we feed into an LTI system: eigenfunction eigenvalue eigenfunction, eigenvalue.

4. Cont.. • describes the change in complex amplitude of a complex exponential input signal as a function of the frequency . • The eigenvalue is called the frequency response (transfer function)of the system, it is the DTFT of . • is a continuous complex function of the frequency . • Specifies amplitude and phase change of the input • It can be expressed in the rectangular form as: Where is the real part, And is the imaginary part. • Or it can be expressed in the polar form as: Where is the magnitude response, And is the phase response.

5. Cont.. • Example: Frequency response of the ideal delay system • Method 1: using the eigenfunction • Method 2: using the impulse response

6. Cont.. • When is small at a frequency , the component at this frequency is essentially removed from the input signal. For this reason, LTI systems are often called Filters. • Different from continuous-time frequency response • Discrete-time frequency response is periodic with period • Then frequencies and are indistinguishable, it follows that we need only specify over an interval of length , e.g., or . • For simplicity and for consistency with the continuous-time case, it is generally convenient to specify over the interval . Then the low frequencies are those that are close to zero, and high frequencies are those that are close to .

7. Example: • Moving Average Then And

8. Cont..

9. Cont.. • The frequency response defines a systems output • for an input as complex exponential at all frequencies • If input signals can be represented as a sum of complex exponentials • we can determine the output of the system • Response to sinusoidal sequences: The response to a sinusoidal input is sinusoidalwith the same frequency and with amplitude and phase determined by the system

10. Cont.. If real, then

11. Frequency response from difference equations • When an LTI system is represented by the difference equation: • We know that when , then must be Then:

12. Cont.. • Example: Consider a stable system described by With We feed the system with to obtain: Since , it follows that And Moreover, if we choose then

13. Cont.. • The normalized gain at is about 0.58 while the phase shift is about radians (or samples).

14. Ideal Filters • An ideal frequency-selective filter has a distortionless response over one or more frequency bands and zero response at the remaining frequencies. • For example, an ideal bandpass Filter (BPF) is defined by: Where and • The parameters and , which specify the end points of the passband, are called the lower and uppercutofffrequencies. • The bandwidth of the filter, defined as the width of the passbandat the positive part of the frequency axis, is given by • An ideal lowpass filter (LPF) is defined with , whereas an ideal highpass (HPF) filter has . • Ideal bandstop (BSF) filters have a distortionless response over all except the stopband, where .

15. Cont..

16. Ideal lowpass filter • The frequency response of an ideal LPF is given by and its corresponding impulse response is defined as • We note that extends from to ; therefore we cannot compute the output of the ideal lowpass filter using a convolution sum. • Moreover, is not absolutely summable, and hence the ideal lowpass filter is (strictly speaking) unstable. • We deduce that we cannot compute the output of an ideal lowpass filter either recursively or non-recursively.

17. Cont.. • The impulse response of the ideal BPF can be obtained by modulating the impulse response of an ideal LPF with using a carrier with frequency . The result is • The impulse responses of the ideal HPF and BSF are given by • Therefore, all ideal filters are unstable and unrealizable. • Ideal filters are usually used in early stages of filter design process to specify the modules in a signal processing system. However, since they are not realizable in practice, they must be approximated by practical or non-ideal filters.

18. Suddenly applied complex Exponential inputs • When we applied as an input The output was • Now if the applied input is of the form i.e., a complex exponential that is suddenly applied at . The output for a causal LTI system with impulse response will be: Since then

19. Cont.. • For , the output can be written as: Then consists of two terms: Where is the steady state response, it is the output when the input is . And is the transient response since in some cases it may become zero. • If the system is FIR, i.e., except for , then for . • If the system is IIR, the transient response does not disappears abruptly, but if the samples of approaches zero with increasing , then will approach zero.

20. Cont.. If , i.e., the system is stable then the transient response must become increasingly smaller as . So a sufficient condition for the transient response to decay is that the system is stable. • The condition for stability is also a sufficient condition for the existence of the frequency response. So the condition Ensures that exists.