1. Lecture 1 Matlab Exercise
Lee-Kang Lester Liu

2. Problem M2.1
M2.1 : write a Matlab Program to generate the conjugate-symmetric and conjugate-anti-symmetric parts of a finite-length complex sequence.

3. Problem M2.1
M2.1 : write a Matlab Program to generate the conjugate-symmetric and conjugate-anti-symmetric parts of a finite-length complex sequence.
What are conjugate-symmetric and conjugate-anti-symmetric ?

4. Problem M2.1
M2.1 : write a Matlab Program to generate the conjugate-symmetric and conjugate-anti-symmetric parts of a finite-length complex sequence.
What are conjugate-symmetric and conjugate-anti-symmetric ?
Conjugate-symmetric :
𝑥 𝑒 𝑛 = 𝑥 𝑒 ∗ −𝑛
Conjugate-anti-symmetric :
𝑥 𝑜 𝑛 = −𝑥 𝑜 ∗ −𝑛
Where * denotes complex conjugate.

5. Problem M2.1
M2.1 : write a Matlab Program to generate the conjugate-symmetric and conjugate-anti-symmetric parts of a finite-length complex sequence.
Any sequence 𝑥 𝑛 can be expressed as a sum of conjugate-symmetric and conjugate-anti-symmetric sequences.

6. Problem M2.1
M2.1 : write a Matlab Program to generate the conjugate-symmetric and conjugate-anti-symmetric parts of a finite-length complex sequence.
Any sequence 𝑥 𝑛 can be expressed as a sum of conjugate-symmetric and conjugate-anti-symmetric sequences.
𝑥 𝑛 = 𝑥 𝑒 𝑛 + 𝑥 𝑜 𝑛

7. Problem M2.1
M2.1 : write a Matlab Program to generate the conjugate-symmetric and conjugate-anti-symmetric parts of a finite-length complex sequence.
Any sequence 𝑥 𝑛 can be expressed as a sum of conjugate-symmetric and conjugate-anti-symmetric sequences.
𝑥 𝑛 = 𝑥 𝑒 𝑛 + 𝑥 𝑜 𝑛
Therefore
𝑥 𝑒 𝑛 = 1 2 𝑥 𝑛 + 𝑥 ∗ −𝑛 = 𝑥 𝑒 ∗ −𝑛
Conjugate-symmetric

8. Problem M2.1
M2.1 : write a Matlab Program to generate the conjugate-symmetric and conjugate-anti-symmetric parts of a finite-length complex sequence.
Any sequence 𝑥 𝑛 can be expressed as a sum of conjugate-symmetric and conjugate-anti-symmetric sequences.
𝑥 𝑛 = 𝑥 𝑒 𝑛 + 𝑥 𝑜 𝑛
Therefore
𝑥 𝑒 𝑛 = 1 2 𝑥 𝑛 + 𝑥 ∗ −𝑛 = 𝑥 𝑒 ∗ −𝑛
Conjugate-symmetric
𝑥 𝑜 𝑛 = 1 2 𝑥 𝑛 − 𝑥 ∗ −𝑛 =− 𝑥 𝑜 ∗ −𝑛
Conjugate-anti-symmetric

9. Problem M2.1
M2.1 : write a Matlab Program to generate the conjugate-symmetric and conjugate-anti-symmetric parts of a finite-length complex sequence.
Matlab Exercise
Given
x[n] = 2𝛿 𝑛 + 3−𝑖 𝛿 𝑛− 𝑖 𝛿 𝑛− 𝑖 𝛿 𝑛−3
+ −2+3𝑖 𝛿 𝑛−4 +5𝛿 𝑛−5

10. Question ?

11. Problem M2.2
M2.2 :
(a) Using program 2_2, generate the sequence shown in Figure 2.23 and 2.24.
(b) Generate and plot the complex exponential sequence −2.7 𝑒 −0.4+ 𝑗𝜋 6 𝑛 for
0≤𝑛≤82 using program 2_2.

12. Problem M2.2(a)
M2.2 :
(a) Using program 2_2, generate the sequence shown in Figure 2.23 and 2.24.
(b) Generate and plot the complex exponential sequence −2.7 𝑒 −0.4+ 𝑗𝜋 6 𝑛 for
0≤𝑛≤82 using program 2_2.

13. Problem M2.2(a)
M2.2 :
(a) Using program 2_2, generate the sequence shown in Figure 2.23 and 2.24.
(b) Generate and plot the complex exponential sequence −2.7 𝑒 −0.4+ 𝑗𝜋 6 𝑛 for
0≤𝑛≤82 using program 2_2.
Index from 0 to 40
A sequence x[n]
= 𝑒 − 𝑗𝜋 6 𝑛

14. Problem M2.2(a)
M2.2 :
(a) Using program 2_2, generate the sequence shown in Figure 2.23 and 2.24.
(b) Generate and plot the complex exponential sequence −2.7 𝑒 −0.4+ 𝑗𝜋 6 𝑛 for
0≤𝑛≤82 using program 2_2.

15. Problem M2.2(a)
M2.2 :
(a) Using program 2_2, generate the sequence shown in Figure 2.23 and 2.24.
(b) Generate and plot the complex exponential sequence −2.7 𝑒 −0.4+ 𝑗𝜋 6 𝑛 for
0≤𝑛≤82 using program 2_2.
Index from 0 to 30
𝑥 𝑛 = 𝑛 3.
𝑥 𝑛 = 𝑛

16. Problem M2.2(b)
M2.2 :
(a) Using program 2_2, generate the sequence shown in Figure 2.23 and 2.24.
(b) Generate and plot the complex exponential sequence −2.7 𝑒 −0.4+ 𝑗𝜋 6 𝑛 for
0≤𝑛≤82 using program 2_2.

17. Question ?

18. Problem M2.5
M2.5 :
Using Matlab to verify the result of Example 2.15.

19. Problem M2.5 M2.5 : Using Matlab to verify the result of Example 2.15.
Consider three sequence generated by uniformly sampling the three cosine functions of frequencies 3Hz, 7Hz, 13Hz, respectively:
𝑔 1 𝑡 =𝑐𝑜𝑠 6𝜋𝑡 𝑠𝑒𝑐 , 𝑔 2 𝑡 =𝑐𝑜𝑠 14𝜋𝑡 𝑠𝑒𝑐 , 𝑔 3 𝑡 =cos⁡ 26𝜋𝑡 𝑠𝑒𝑐
with sampling rate 𝐹 𝑇 =10𝐻𝑧 , that is , 𝑇=0.1(sec⁡)

20. Problem M2.5 M2.5 : Using Matlab to verify the result of Example 2.15.
Consider three sequence generated by uniformly sampling the three cosine functions of frequencies 3Hz, 7Hz, 13Hz, respectively:
𝑔 1 𝑡 =𝑐𝑜𝑠 6𝜋𝑡 𝑠𝑒𝑐 , 𝑔 2 𝑡 =𝑐𝑜𝑠 14𝜋𝑡 𝑠𝑒𝑐 , 𝑔 3 𝑡 =cos⁡ 26𝜋𝑡 𝑠𝑒𝑐
with sampling rate 𝐹 𝑇 =10𝐻𝑧 , that is , 𝑇=0.1(sec⁡)
Using Eq 2.63 and Eq 2.63
𝑥 𝑛 =𝐴𝑐𝑜𝑠 2𝜋 𝑓 0 𝑛𝑇

21. Problem M2.5 M2.5 : Using Matlab to verify the result of Example 2.15.
Consider three sequence generated by uniformly sampling the three cosine functions of frequencies 3Hz, 7Hz, 13Hz, respectively:
𝑔 1 𝑡 =𝑐𝑜𝑠 6𝜋𝑡 𝑠𝑒𝑐 , 𝑔 2 𝑡 =𝑐𝑜𝑠 14𝜋𝑡 𝑠𝑒𝑐 , 𝑔 3 𝑡 =cos⁡ 26𝜋𝑡 𝑠𝑒𝑐
with sampling rate 𝐹 𝑇 =10𝐻𝑧 , that is , 𝑇=0.1(sec⁡)
Using Eq 2.63 and Eq 2.63
𝑥 1 𝑛 =𝑐𝑜𝑠 0.6𝜋𝑛+2𝜋𝑘
𝑥 2 𝑛 =𝑐𝑜𝑠 1.4𝜋𝑛+2𝜋𝑘
𝑥 3 𝑛 =𝑐𝑜𝑠 2.6𝜋𝑛+2𝜋𝑘

22. Problem M2.5 M2.5 : Using Matlab to verify the result of Example 2.15.
Consider three sequence generated by uniformly sampling the three cosine functions of frequencies 3Hz, 7Hz, 13Hz, respectively:
𝑔 1 𝑡 =𝑐𝑜𝑠 6𝜋𝑡 𝑠𝑒𝑐 , 𝑔 2 𝑡 =𝑐𝑜𝑠 14𝜋𝑡 𝑠𝑒𝑐 , 𝑔 3 𝑡 =cos⁡ 26𝜋𝑡 𝑠𝑒𝑐
with sampling rate 𝐹 𝑇 =10𝐻𝑧 , that is , 𝑇=0.1(sec⁡)
Using Eq 2.63 and Eq 2.63
𝑥 1 𝑛 =𝑐𝑜𝑠 0.6𝜋𝑛+2𝜋𝑘 = cos 0.6𝜋𝑛 where 𝑘=0
𝑥 2 𝑛 =𝑐𝑜𝑠 1.4𝜋𝑛+2𝜋𝑘 =𝑐𝑜𝑠 −0.6𝜋𝑛 where 𝑘=−1
𝑥 3 𝑛 =𝑐𝑜𝑠 2.6𝜋𝑛+2𝜋𝑘 =𝑐𝑜𝑠 0.6𝜋𝑛 where 𝑘=−1

23. Question ?

24. Problem M3.1
M3.1 :
Determine and plot the real and imaginary parts and the magnitude and phase spectra of the following DTFT for various value of 𝛾 and 𝜃.
𝐺 𝑒 𝑗𝜔 = 1 1−2𝛾 𝑐𝑜𝑠𝜃 𝑒 −𝑗𝜔 + 𝛾 2 𝑒 −2𝑗𝜔
0<𝛾<1

25. Problem M3.1
M3.1 :
Determine and plot the real and imaginary parts and the magnitude and phase spectra of the following DTFT for various value of 𝛾 and 𝜃.
𝐺 𝑒 𝑗𝜔 = 1 1−2𝛾 𝑐𝑜𝑠𝜃 𝑒 −𝑗𝜔 + 𝛾 2 𝑒 −2𝑗𝜔
0<𝛾<1
In z-plane , what are those roots in denominator ?
𝐺 𝑧 = 1 1−2𝛾 𝑐𝑜𝑠𝜃 𝑧 −1 + 𝛾 2 𝑧 −2
0<𝛾<1

26. Problem M3.1
M3.1 :
Determine and plot the real and imaginary parts and the magnitude and phase spectra of the following DTFT for various value of 𝛾 and 𝜃.
𝑧= 2𝛾 𝑐𝑜𝑠𝜃 ± −2𝛾 𝑐𝑜𝑠𝜃 2 −4 𝛾 2 2
= 2𝛾 𝑐𝑜𝑠𝜃 ± 4 𝛾 2 𝑐𝑜𝑠 2 𝜃 −1 2
= 𝛾𝑐𝑜𝑠𝜃±𝛾𝑠𝑖𝑛𝜃 −1
=𝛾 𝑐𝑜𝑠𝜃±𝑗𝑠𝑖𝑛𝜃
0<𝛾<1

27. Problem M3.1
M3.1 :
Determine and plot the real and imaginary parts and the magnitude and phase spectra of the following DTFT for various value of 𝛾 and 𝜃.
𝑧= 2𝛾 𝑐𝑜𝑠𝜃 ± −2𝛾 𝑐𝑜𝑠𝜃 2 −4 𝛾 2 2
= 2𝛾 𝑐𝑜𝑠𝜃 ± 4 𝛾 2 𝑐𝑜𝑠 2 𝜃 −1 2
= 𝛾𝑐𝑜𝑠𝜃±𝛾𝑠𝑖𝑛𝜃 −1
=𝛾 𝑐𝑜𝑠𝜃±𝑗𝑠𝑖𝑛𝜃
0<𝛾<1
Note : these roots are poles of the transfer function.

28. Question ?

29. Problem M3.3(b)
M3.1 :
Determine and plot the real and imaginary parts and the magnitude and phase spectra of the following DTFT.
X 𝑒 𝑗𝜔 = 𝑒 −𝑗𝜔 − 𝑒 −𝑗2𝜔 − 𝑒 −𝑗3𝜔 𝑒 −𝑗𝜔 𝑒 −𝑗2𝜔 − 𝑒 −𝑗3𝜔

30. Problem M3.3(b)
M3.1 :
Determine and plot the real and imaginary parts and the magnitude and phase spectra of the following DTFT.
X 𝑒 𝑗𝜔 = 𝑒 −𝑗𝜔 − 𝑒 −𝑗2𝜔 − 𝑒 −𝑗3𝜔 𝑒 −𝑗𝜔 𝑒 −𝑗2𝜔 − 𝑒 −𝑗3𝜔
Using Matlab roots function to check its zeros.!!

31. Question ?