1. Lect3 Frequency Domain Analysis (Discrete Fourier Series and the Fourier Transform)

2. 3.1 Introduction
Why do we need also a frequency domain analysis (also we need time domain convolution):-
1) Sinusoidal and exponential signals occur in the natural word, and in the world technology.
2) If an input signal is described by its frequency spectrum and an LTI system by its frequency response, then the output signal spectrum is found by multiplication both.
3) The design of DSP algorithms and systems often starts with a frequency domain specification.

3. 3.2 Discrete Fourier Series
Let us consider a periodic digital signal x[n] as shown in fig(3.1), such as
where N is the period of the sequence x[n].
The coefficients of its line spectrum indicate the amount of various frequencies contained in the signal.

5. 3.3 Power and Energy of periodic signal
Parceval's theorem can be used to calculate the total power or energy of a signal in the time and frequency domains

6. 3.4 Properties of the Fourier Series

7. 3.5 The Fourier Transform of Aperiodic Digital Sequences

8. What happens to the spectral coefficients 𝐶𝑘 if we stretch the signal in this way:-
(a) Coefficient of 𝐶𝑘 become smaller because of the 1/N multiplier in equation (3.12).
(b)Coefficient of 𝐶𝑘 becomes closer in frequency because N also appears in the denominator of the exponential.

9. 3.6 Properties of Fourier Transfers of the aperiodic digital

10. 3.7 Frequency Response of LTI Processors
3.7.1 The first method by using input signal frequency response and LTI frequency response
The key relationships defining an LTI system in the time domain and frequency domain are summarized by fig (3.3)

11. 3.7.2 LTI Systems Characterized by Linear Constant Coefficients Difference Equation (LCCDE) [2nd method to fined frequency response]
In this section we focus on a family of LTI systems described by input-output relation called a difference equation with constant coefficients.
Fig(3.4) shows a block diagram of a simple recursive systemdescribed by a first order difference equation as follows

12. There are two method for this solution:-
3.7.3 Solution of Linear Constant Coefficient Difference Equation (LCCDE). (Third method to find frequency response)
There are two method for this solution:-
1) Direct method.
2) Indirect method based on Z-transform

13. 3.8The Z-transform and its applications in signal processing [Fourth method to find frequency response]
3.8.1 The Z-transform
The Z-transform of a sequence, x[n], which is valid for all n
3.8.2 The Inverse Z-transform
- The inverse z-transform (IZT) allows recovering the discrete time sequence, x[n], given its transform. The IZT is particularly useful in DSP, for example in finding the impulse response of digital filters. The inverse z-transform is defend as

14. Practice, X(z) is often expressed as a ratio of two polynomials in 𝑧−1
In this form the inverse-z transform, x[n] can be obtained using one of several methods:-
1) Power series expansion method
2) Partial fraction expansion method
3) Residue
Power series method
Partial fraction expansion method
3.8.3 Properties of the z-transform

15. 3.8.4 Pole-Zero description of discrete –time system
- In most practical discrete- time systems the z-transform of the transfer function H(z), can be expressed in terms of poles and zeros.
If H(z) has poles at 𝑧=𝑃1,𝑃2,… 𝑃𝑁
And zero at 𝑧=𝑧1,𝑧2,… 𝑧𝑁. The H(z) can be factored and represents as

16. 3.8.5 Frequency response estimation
Geometric evaluation of frequency response
If the z-transform of an LTI system is expressed as

17. 3.8.5.2 Frequency response estimation via FFT
FFT may also be used to evaluate the frequency response of discrete time systems.
3.8.6 Stability Consideration
Stability analysis is often carried out as part of the design of discrete-time systems.