# 代写信号分析 – Topics on Data and Signal Analysis

### Topics on Data and Signal Analysis ### Coursework I

Problem 1.Consider the following three vectors inR^3 :

`````` 1 = [11 2]T,  2 = [2 32]T,  2 = [3 1 1]T.
``````
``````a) Does the set  ={ 1 , 2 , 3 }form a basis forR^3?
``````
``````b) If  ={ 1 , 2 , 3 }forms a basis, find its biorthogonal basis  ={ 1 , 2 , 3 }.
``````
``````c) For an arbitraryx= [x 1 x 2 x 3 ]TinR^3 describe the procedure and the expression for finding coefficients
c 1 ,c 2 andc 3 such that
x=c 1  1 +c 2  2 +c 3  3.
``````
``````d) Find the largest numberA>0 and the smallest numberB<such that
``````
``````Ax^2
``````

#### ^3

``````i=
``````
``````|i,x|^2 Bx^2
``````
``````for allxR^3.
``````

Problem 2.Consider the systems in Figures 1-4.

`````` 2 3   3 2
``````
`````` 2 4   4 2
``````
`````` 1  2
``````
`````` 3  4
``````
``````a) For the signalx[n] which is given in the Fourier domain by the figure below, sketchY(ej) for each of the
four systems in the above.
``````
``````!"
``````
`````` 3   2   #  3
``````
``````1
``````
``````#  2
``````
``````!"
``````
``````
``````
``````
2
``````
``````
2
``````
``````1
``````
``````b) For any arbitrary signalx[n]  express samples ofy[n] in terms of samples ofx[n] for each of the four
systems in the above.
``````

Problem 3.Consider a filterh[n].

``````a) Find the Fourier transform of its autocorrelation sequence
``````
``````a[n] =
``````
``````k
``````
``````h[k]h[kn].
``````
``````b) Show that ifh[kn],h[k]=[n], then
``````
``````|H(ej)|= 1,.
``````
``````c) Show that if|H(ej)|= 1,then
``````
``````h[kn],h[k]=[n].
``````
``````d) Show that if|H(ej)|= 1,,then{h[kn],nZ}is an orthonormal basis for`^2 (Z).
``````

Problem 4.Consider two waveforms 0 [n] and 1 [n] and two waveform 0 [n] and 1 [n] in`^2 (Z). Leth 0 [n] and h 1 [n] be two filters such thath 0 [n] = 0 [n] andh 1 [n] = 1 [n], andg 0 [n] andg 1 [n] two filters such that g 0 [n] = 0 [n] andg 1 [n] = 1 [n].

``````a) Show that if 0 [n], 0 [n 2 k]=[k] thenH 0 (z)G 0 (z) +H 0 (z)G 0 (z) = 2 and that if
1 [n], 1 [n 2 k]=[k] thenH 1 (z)G 1 (z) +H 1 (z)G 1 (z) = 2.
``````
``````b) Show that if 0 [n], 1 [n 2 k]= 0 for allkZthenH 1 (z)G 0 (z) +H 1 (z)G 0 (z) = 0 and that if
1 [n], 0 [n 2 k]= 0 for allkZthenH 0 (z)G 1 (z) +H 0 (z)G 1 (z) = 0.
``````
``````c) Using the results of a) and b) show that if 0 [n], 0 [n 2 k]=[k] , 1 [n], 1 [n 2 k]=[k],
0 [n], 1 [n 2 k]= 0 for allkZ, and 1 [n], 0 [n 2 k]= 0 for allkZ, then
``````
``````H 0 (z)G 0 (z) +H 1 (z)G 1 (z) = 2andH 0 (z)G 0 (z) +H 1 (z)G 1 (z) = 0.
``````