◎ This article is downloaded from the Internet ： Signals and systems 2022 Spring term homework - The ninth assignment : https://blog.csdn.net/zhuoqingjoking97298/article/details/124319518

 

§01 The base Basic work

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1.1 Signal sampling and recovery

1.1.1 Sampled signal spectrum

   Let the bottom width of triangle and raised cosine signal be $\tau$ , The sampling interval is $T_s=\tau /8$ when , Draw the spectrum of the sampled signal respectively .

▲ chart 1.1.1 Triangle and rising cosine signal

1.1.2 Rectangular signal sampling frequency

   The bottom width of the rectangular signal is known $\tau =2.5ms$ . Let the spectrum of the signal be 8 The spectral components beyond the zero crossing points can be ignored , What is the minimum sampling frequency for this model ？

▲ chart 1.1.2 Signal and corresponding spectrum

1.2 Laplace transform and z Transformation

1.2.1 Find the Laplace transform of the signal

   Find the Laplace transform of the following functions , And draw the zero pole distribution diagram of the signal . By default , Is to find the unilateral Laplace transform of the signal .

$$\left(1\right)1-e^{-at};\left(2\right)\sin t+2\cos t;$$$$\left(3\right)t\cdot e^{-2t};\left(4\right)e^{-t}\sin 2t;$$$$\left(5\right)\left(1+2t\right)\cdot e^{-t};\left(6\right)\left(1-\cos \alpha t\right)e^{-\beta t};$$$$\left(7\right)t^2+2t;\left(8\right)2\delta \left(t\right)-3e^{-7t};$$$$\left(9\right)e^{-\alpha t}\sinh \left(\beta t\right);\left(10\right){\cos }^2\left(\Omega t\right);$$$$\left(11\right)\frac{1}{\beta -\alpha }\left(e^{-\alpha t}-e^{-\beta t}\right);\left(12\right)e^{-\left(t+a\right)}\cos \omega t;$$$$\left(13\right)t\cdot e^{-\left(t-2\right)}\cdot u\left(t-1\right);\left(14\right)e^{-\frac{t}{a}}f\left(\frac{t}{a}\right)$$$$\left(15\right)e^{-at}f\left(\frac{t}{a}\right);\left(16\right)t\cdot {\cos }^3\left(3t\right);$$$$\left(17\right)t^2\cdot \cos \left(2t\right);\left(18\right)\frac{1}{t}\left(1-e^{-at}\right);$$$$\left(19\right)\frac{e^{-3t}-e^{-5t}}{t};\left(20\right)\frac{\sin \left(\alpha t\right)}{t};$$

   among , If you include functions $f\left(t\right)$ , So suppose it corresponds to Laplace Transformation for $F\left(s\right)$ .

（1） It must be done

   The above signals have even numbers .

   Of other signals Laplace Transformations are all Choose a question .

1.2.2 Signal seeking z Transformation

   Find the of the following sequence signals $z$ Transformation $X\left(z\right)$ , It is shown that the convergence region , Will come out $X\left(z\right)$ Zero pole diagram of .

$$\left(1\right){\left(\frac{1}{4}\right)}^{n}u\left(t\right);\left(2\right){\left(-\frac{1}{3}\right)}^{n}u\left(t\right);$$$$\left(3\right){\left(\frac{1}{2}\right)}^{n}u\left(t\right);\left(4\right){\left(-\frac{1}{2}\right)}^{n}u\left[-n\right];$$$$\left(5\right)-{\left(\frac{1}{6}\right)}^{n-1}u\left[-n-1\right];\left(6\right)u\left[n-1\right];$$ ( 7 )      ( 1 10 ) n { u [ n ] − u [ n − 3 ] } ; \left( 7 \right)\,\,\,\,\left( { {1 \over {10}}} \right)^n \left\{ {u\left[ n \right] - u\left[ {n - 3} \right]} \right\}; (7)(101​)n{ u[n]−u[n−3]};$$\left(8\right){\left(\frac{1}{5}\right)}^{n}u\left[n\right]+{\left(\frac{1}{6}\right)}^{n}u\left[n\right];$$$$\left(9\right)\delta \left[n\right]-\frac{1}{2}\delta \left[n-8\right]$$

（1） It must be done

   The above signals have even numbers .

   Of other signals z Transformations are all “ Choose a question ”.

1.2.3 Bilateral sequence z Transformation

   Qiushi Xiaobian sequence $$x\left[n\right]$$ Of $z$ Transformation , And indicate the convergence region . Draw out $z$ Pole zero diagram of transformation .

$$x\left[n\right]={\left(\frac{1}{a}\right)}^{|n|},\left|a\right|>1$$

 

§02 real Inspection operation

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2.1 MATLAB Medium Laplace,ZT Transformation of

   stay MATLAB Practice using laplace 、 ilaplace 、 ztrans 、 iztrans Command complete Laplace Transform sum z Transformation and inverse transformation . And verify the corresponding results in the previous system .

2.1.1 MATLAB To solve the problem LT Code

（1）Laplace Transformation

   test Laplace Transformation example ：
  （1） f $f\left(t\right)=t^2$ ;

  （2） $f\left(t\right)=e^{-3t}$

>> syms t
>> laplace(t^2)
ans = 2/s^3
>> laplace(exp(4*t))
ans = 1/(s-4

（2）Laplace Reverse transformation

   measurement Laplace Inverse transformation example ：

$$\left(1\right)\frac{1}{{\left(s+1\right)}^2};\left(2\right)\frac{1}{s^2+4};$$

>> syms s
>> ilaplace(1/(1+s)^2)
ans = t*exp(-t)
>> ilaplace(1/(s^2+4))
ans = 1/4*4^(1/2)*sin(4^(1/2)*t)

2.1.2 MATLAB To solve the problem z Transform code

   Here's the code for the test ：

ztrans(f)
>> f=n^4
>> ztrans(f)
>ans = z*(z^3+11*z^2+11*z+1)/(z-1)^5

iztrans(f)
>> iztrans(z/(z-2))
ans = 2^n

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■ Links to related literature :

● Related chart Links :

• chart 1.1.1 Triangle and rising cosine signal
• chart 1.1.2 Signal and corresponding spectrum