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Lecture 1 Matlab Exercise
Lee-Kang Lester Liu
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.
Problem M2.1
What are conjugate-symmetric and conjugate-anti-symmetric ?
M2.1 : write a Matlab Program to generate the conjugate-symmetric and conjugate-anti-symmetric parts of a finite-length complex sequence.
Problem M2.1
𝑥𝑒❑ [𝑛 ]=𝑥𝑒
∗ [−𝑛 ]Conjugate-symmetric : Conjugate-anti-symmetric :
𝑥𝑜❑ [𝑛 ]=−𝑥𝑜
∗ [−𝑛 ]
What are conjugate-symmetric and conjugate-anti-symmetric ?
Where * denotes complex conjugate.
M2.1 : write a Matlab Program to generate the conjugate-symmetric and conjugate-anti-symmetric parts of a finite-length complex sequence.
Problem M2.1
Any sequence can be expressed as a sum of conjugate-symmetric and conjugate-anti-symmetric sequences.
M2.1 : write a Matlab Program to generate the conjugate-symmetric and conjugate-anti-symmetric parts of a finite-length complex sequence.
Problem M2.1
Any sequence can be expressed as a sum of conjugate-symmetric and conjugate-anti-symmetric sequences.
𝑥 [𝑛 ]=𝑥𝑒❑ [𝑛 ]+𝑥𝑜
❑ [𝑛 ]
M2.1 : write a Matlab Program to generate the conjugate-symmetric and conjugate-anti-symmetric parts of a finite-length complex sequence.
Problem M2.1
Any sequence can be expressed as a sum of conjugate-symmetric and conjugate-anti-symmetric sequences.
𝑥 [𝑛 ]=𝑥𝑒❑ [𝑛 ]+𝑥𝑜
❑ [𝑛 ]Therefore 𝑥𝑒
❑ [𝑛 ]=12
(𝑥 [𝑛 ]+𝑥∗ [−𝑛 ] )=𝑥𝑒∗ [−𝑛 ] Conjugate-
symmetric
M2.1 : write a Matlab Program to generate the conjugate-symmetric and conjugate-anti-symmetric parts of a finite-length complex sequence.
Problem M2.1
Any sequence can be expressed as a sum of conjugate-symmetric and conjugate-anti-symmetric sequences.
𝑥 [𝑛 ]=𝑥𝑒❑ [𝑛 ]+𝑥𝑜
❑ [𝑛 ]Therefore 𝑥𝑒
❑ [𝑛 ]=12
(𝑥 [𝑛 ]+𝑥∗ [−𝑛 ] )=𝑥𝑒∗ [−𝑛 ]
𝑥𝑜❑ [𝑛 ]=1
2(𝑥 [𝑛 ]−𝑥∗ [−𝑛 ] )=−𝑥𝑜
∗ [−𝑛 ] Conjugate-anti-symmetric
Conjugate-symmetric
M2.1 : write a Matlab Program to generate the conjugate-symmetric and conjugate-anti-symmetric parts of a finite-length complex sequence.
Problem M2.1
Matlab Exercise
Given x[n] =
M2.1 : write a Matlab Program to generate the conjugate-symmetric and conjugate-anti-symmetric parts of a finite-length complex sequence.
Question ?
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 for using program 2_2.
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 for using program 2_2.
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 for using program 2_2.
1. Index from 0 to 40
2. A sequence x[n] =
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 for using program 2_2.
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 for using program 2_2.
1. Index from 0 to 30
2.
3.
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 for using program 2_2.
Question ?
Problem M2.5
M2.5 : Using Matlab to verify the result of Example 2.15.
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:
with sampling rate , that is ,
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:
with sampling rate , that is ,
Using Eq 2.63 and Eq 2.63
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:
with sampling rate , that is ,
Using Eq 2.63 and Eq 2.63
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:
with sampling rate , that is ,
Using Eq 2.63 and Eq 2.63 where where where
Question ?
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 and .
𝐺 (𝑒 𝑗 𝜔 )= 1
1−2𝛾 (𝑐𝑜𝑠𝜃 )𝑒− 𝑗 𝜔+𝛾 2𝑒−2 𝑗 𝜔0<𝛾<1
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 and .
𝐺 (𝑒 𝑗 𝜔 )= 1
1−2𝛾 (𝑐𝑜𝑠𝜃 )𝑒− 𝑗 𝜔+𝛾 2𝑒−2 𝑗 𝜔0<𝛾<1
In z-plane , what are those roots in denominator ?
𝐺 (𝑧 )= 1
1−2𝛾 (𝑐𝑜𝑠𝜃 ) 𝑧−1+𝛾 2𝑧−20<𝛾<1
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 and .
0<𝛾<1
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 and .
0<𝛾<1
Note : these roots are poles of the transfer function.
Question ?
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
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
Using Matlab roots function to check its zeros.!!
Question ?