1. Review of Fourier Transform for Continuous-time aperiodic signals.

2. Summary of the approach to obtain the Fourier Transform for Discrete-time aperiodic signals.

3. Development of Discrete-time Fourier Transforms. DTFT.
x[n}
-N1 N2 -N
-N1 N2 N

4. Development of Discrete-time Fourier Transforms.

5. Development of Discrete-time Fourier Transforms.

6. Development of Discrete-time Fourier Transforms.

7. Development of Discrete-time Fourier Transforms.

8. Discrete-time Fourier Transform

9. Example 5.1 Discrete-time Fourier Transform.

10. Example 5.1 Discrete-time Fourier Transform.
|X(w)| w -p 2p p
Phase X(w) -p p 2p

11. Discrete-time Fourier Transform for rectangular pulse
Example 5.3 | 1
x[n] -2 2 n
X(w) 5 w -p p

12. X(w) w -p p

13. Review of relationships for Fourier transform associated with continuous-time periodic signals

14. Summary of relationships for DT Fourier transform associated with DT periodic sequences

15. -N
-N1 N2 N

16. Illustration of relationship of DTFS & DTFT
-N1 N1 N n
Case N=20, 2N1+1=5.
1/4
1/4 | 5 6 7 8 12 1 2 3 4 9 10 11 -1 19 20 -2
x2p/20

17. Summary of relationships for DT Fourier transform associated with DT periodic sequences

18. Discrete-time Fourier Transforms. DTFT.
The synthesis equation for the discrete sequence x[n] (which is aperiodic in nature ) is made up of the linear combination of the basic complex exponential signal. The combination here is through the weights of X(jw) and the integral combination over an interval period of 2 pi radians.
The analysis equation is the equation that covert the sequence in the time domain to the frequency. Notice that although the time domain signal or sequence is discrete in nature, the frequency domain is not discrete but tends to be continuous in frequencies. The spectrum X(w) is complex and it has the real part and the imaginary part. Or in the polar coordinate form it has the magnitude and the phase/angle.

19. Properties of Discrete-time Fourier Transforms.

20. Properties of Discrete-time Fourier Transforms.

21. Properties of Discrete-time Fourier Transforms.

22. Properties of Discrete-time Fourier Transforms.

23. Convolution Property
x[n]
h[n]
H(w)
h[n]*x[n]
X(w)
H(w)X(w)

24. Filtering
Changing the relative amplitudes of the frequency components.
Linear time-invariant systems that change the shape of the spectrum are referred to as frequency-shaping filters.
Systems that pass some frequencies and attenuate or eliminate others are referred to as frequency-selective filters.

25. Frequency-shaping Filters
Audio Systems
LTI filters allows the listeners to modify the relative amounts of low-frequency energy(bass) and high frequency energy (treble).
Frequency response of these filters are changed via manipulating the tone control of your Hi-Fi system.

26. Frequency-shaping Filters
Also in Hi-Fi Audio Systems we will have equalizing filters included in the preamplifier to compensate for the frequency response characteristics of the speakers.
Overall these cascaded filtering stages are frequently referred to as the equalizing or equalizer circuits for the audio system.

27. Magnitude Frequency Plot of the Frequency response of LTI Systems.
Bode Plot
20log10|H(w)| in decibels (dB)
0db
100
1000 10
Frequency in logarithmic scale

28. Phase Frequency Plot of the Frequency response of LTI Systems.
Bode Plot
Angle H(jw)
P/2 10
100
1000
-p/2
Frequency in logarithmic scale

29. Magnitude Frequency Plot of the Frequency response of Differentiating Filters.
Enhancing rapid variations or transitions in a signal
e.g. enhance edges in picture processing.
|H(w)|
y(t)=dx(t)/dt
H(w)=Y(w)/X(w)=jw.
Frequency

30. Frequency-Selective Filters
Frequency response of a discrete-time ideal lowpass filter
H1(w) p 2p
-2p -p
-wc wc
Passband w
Stopband
Stopband

31. Frequency response of a discrete-time ideal lowpass & highpass filter
H1(w) 2p
-2p -p
-wc p wc w
H2(w) p -p

32. Frequency response of a discrete-time ideal lowpass & bandpass filter
H1(w)
-2p -p
-wc p 2p wc w
H2(w) p -p