INFE 5201 SIGNALS AND SYSTEMS

Transcription

1 INFE 50 SIGNALS AND SYSTEMS Assignment : Introduction to MATLAB Name, Class&Student ID Aim. To give student an introduction to basic MATLAB concepts. You are required to produce basic program, learn basic calculation and get familiar with MATLAB work environment.. To know how to generate basic signals and functions, such as square and sinc. You are required to produce programs and compare them with each other. Introduction. MATLAB s basic operation ) The best way to get start with MATLAB is to learn how to handle matrices. Usually, MATLAB thinks of everything as a matrix, which is a rectangular array of numbers. To enter a matrix, simply type: A=[ 3; 4 5 6; 7 8 9;] Then, MATLAB will display the matrix you just entered. ) The transpose operation is denoted by an apostrophe or single quote, '. It flips a matrix about its main diagonal and it turns a row vector into a column vector. 3) Concatenation is the process of join small matrices to make bigger ones. The pair of square bracket, [ ], is the concatenation operator. For example, start with the above 3*3 matrix A, B=[A A+], the result B is a 3*6 matrix: B=[ 3 3 4; ; ] 4), and ; are used to indicate the end of a sentence. If you type A=[ 3; 4 5 6; 7 8 9;], or A=[ 3; 4 5 6; 7 8 9;]; MATLAB won t display the matrix you just entered. If you type A=[ 3; 4 5 6; 7 8 9;] MATLAB will display the matrix you just entered. 5) The colon, :, is one of MATLAB s most important operators. It occurs in several different forms. The expression :0 is a row vector containing the integers from to0. To obtain nonunit

2 spacing, specify an increment. For example, 00:-7:50 is ) When you don t specify an output variable, MATLAB uses the variable ans, short for answer, to store the results of a calculation. 7) MATLAB has some operators, such as +, -, *, ^,',( ) Please use help to find out their meanings. 8) MATLAB provides a large number of standard elementary mathematical functions, including abs, sqrt, exp, and sin. MATLAB also provides many more advanced functions, including Bessel and gamma functions. For a list of the elementary mathematical functions, type help elfun. For a list of more advanced mathematical and matrix functions, type help specfun, help elmat. Some of the functions, like sqrt and sin, are built- in. They are part of the MATLAB core so they are very efficient, but the computational details are not readily accessible. Other functions, like gamma and sinh, are implemented in M-files, You can see the code and even modify it if you want. 9) Several special functions provide values of useful constants. For instance, pi is , i is imaginary unit, j is as same as i. 0) The function stem(y) plots the data sequence y as stems from the x axis terminated with circles for the data value. Please turn to help stem for more information. ) The function subplot Create axes in tiled positions. H = subplot (m,n,p), or subplot (mnp), breaks the Figure window into an m-by-n matrix of small axes, selects the p-th axes for the current plot, and returns the axis handle. The axes are counted along the top row of the Figure window, then the second row, etc. For example, subplot(,,), plot(income) subplot(,,),plot(outgo) plots income on the top half of the window and outgo on the bottom half. ) You can use % to give your own comment. MATLAB won t calculate text after %. For example, f=00 %Set frequency (text after % taken as commen 3) Please use help to find out how the functions fix, plot, figure, title, xlable, ylable perform. You can create your own matrices using M-files, which are text files containing MATLAB code. Just create a file containing the same statements you would type at the MATLAB command line. 4) Save the file under a name that ends in.m. To access a text editor on a PC, choose Open or New from the File menu or press the appropriate button on the toolbar.

3 . Two basic signals. ) The function square(t) generates a square wave with period *Pi for the elements of time vector T. square (T) is like sin(t), only it creates a square wave with peaks of + to - instead of a sine wave. ) The sinc function computes the mathematical sinc function for an input vector or matrix x. The sinc function is the continuous inverse Fourier transform of the rectangular pulse of width and height. sinc( = sin( t t 0 t=0 Requirements. Matrix calculation ) Set A=[ 3;4 5 6;7 8 9;]. Please transpose A. ) Set A=[ 3;4 5 6;7 8 9;]. Please calculate [A A*3]. 3) Set A= [ ] B= [ ] Please calculate A+B. 4) Set x=[- 0 ]' y=[- - ]'. Please calculate x *y,y *x and x*y'. 5) Set x=[,0,-]. Please calculate pi*x. 6) Generate 4 3 random matrix.. To generate number series ) x=:5 ) y=0:pi/4:pi 3) x=0:0.: 4) Please describe what did you find out by doing ), ) and 3). 5) x=(0:0.:3.0)'; y=exp(-x).*sin(x); Please produce [x y]. 3. Data calculation ) x=[ 3]; y=[4 5 6]; Please calculate x+y. ) Set A=[ ;3 4] B=[4 3; ]; Please calculate A.^, A.^B, and give theirs meanings respectively in your report. 3) x=[ 3]; y=[4 5 6]; Please calculate x.*y, x.\y and y./x. Please find the

4 difference between.\ and./. Describe it in your report. 4. Math function A=[ 3; 4 5 6] B=fix(pi*A) Please calculate cos(pi*b). 5. Draw picture x=0:.05:4*pi; y=sin(x); Please use plot to draw a picture of x and y. 6. Title Please use title, xlable, ylabel to give titles to the picture you drew just now. 7. To generate some basic signals ) To generate square signal Please generate a 30 H square wave function. The Magnitude value is ; the time interval is 0.00s; the time is from 0 to 0.5s. Please plot the square wave. (a) Use the function square. (b) Refer to help square for help. (c) Set the square function y=(*pi*30*. (d) Write down the result on your report. ) Please generate sinc function sin( y= ; if t=0, y= t (a) Refer to help sinc for help (b) Use figure and plot to show your result. (c) Write down the result on your report. 3) To generate exponential signal Please generate x(n)= e ( 0. j0.) n 0 n 0. Then draw x(n) s real part, imaginary part, magnitude part, and phase part. (a) Use exp to produce exponential signal. (b) Set n=[-0::0], alpha= j; (c) Use subplot and stem to show x(n) s real part, imaginary part, magnitude part, and phase part in a single picture. For example, stem(n,real(x));. (d) Please give each sub-picture a title by using title and xlabel. (e) Analye your results in your report.

5 INFE 50 SIGNALS AND SYSTEMS Assignment : Fourier Transform and Linear System Models Name, Class&Student ID Aim. To review analog signal s FT and digital signal s DFT.. To get system s frequency transform function H(jw), and solve system transform s roots. 3. To get an analog system s frequency response Introduction ) The foundation of the Signal Processing Toolbox is the Fast Fourier Transform (FFT), a method for computing the DFT with reduced execution time. Many of the toolbox functions (including -domain frequency response, some filter design and implementation functions) incorporate the FFT. MATLAB provides the functions fft and ifft to compute the discrete Fourier transform and its inverse, respectively. fft, with a single input argument x, computer the DFT of the input vector or matrix. If x is a vector, fft computers the DFT of the vector; if x is a rectangular array, fft computes the DFT of each array column. For example, create a time vector and signal. t=(0:/99:); % time vector x=sin(*pi*5*+sin(*pi*40*; % signal The DFT of the signal, and the magnitude of the transformed sequence, are then y=fft(x); % Compute DFT of x. m=abs(y); % magnitude A second argument to fft specifies a number of points n for the transform, representing DFT length. y=fft(x,n); In this case, fft pads the input sequence with eros if it is shorter than n, or truncates the sequence if it is longer than n. If n is not specified, it defaults to the length of the input sequence. ) The function axis is used to control axis scaling and appearance. Please turn to help axis for more information. 3) The function roots(c) computes the roots of the polynomial whose coefficients are the elements of the vector c. Please refer to help polyval to find out how to use the function polyval. 4) Frequency Response (Analog Domain): freqs evaluates frequency response for an analog filter by two input coefficient vectors b and a. Its operation is similar to that of freq; you can specify a number of frequency points to use (by default, the function uses 00), supply a vector of arbitrary frequency points, and plot the magnitude and

6 phase response of the filter. 5) The function clf is to used to clear current figure. Another function clear is to clear variables and functions from memory. Don t got confused by them. 6) figure(); figure(); figure(n) can display a total of n pictures in n windows. Requirements. DFT Given j / 6 n (0.9e 0< = n < = 0, 0 j X ( e ) n0 x ( n) ) e jn x( n), please complete the following program DFT for given x(n). Please plot X( magnitude and angle parts frequency outline. (a) If we set x=[x(0) x() x(k)...x(n)], j e ) k=0,,...,n. It will separate a number of (n+) points in [ 0, ] duration as the interval of k / n. j kn m j (b) If W { e,0 n 0, k 0,,, M}, then X X ( e ) WX, where T W [exp( j k n)] M (c) The program is as following. Please complete it. n=0:0; x=(0.9*exp(j*pi/6)).^n; k=-00:00; w=(pi/00)*k; X= ; magx= ; angx= ; figure(); subplot(,,); plot(w/pi,magx); grid; axis([-,,0,0]); xlabel( frequency in units of pi ); ylabel( X ); title( Magnitude Part ); subplot(,,); plot(w/pi,angx/pi); grid; axis([-,,-.5,.5]); xlabel( frequency in units of pi ); ylabel( radians/pi ); title( Phase Part );. FFT Set sampling instants from 0 to 0.6s in steps of ms. Generate a compound sinusoidal sequence x of your own value. Please calculate x s FFT by using fft(x) or fft(x,n). Please plot the magnitude of the transformed sequence (a) n should be power of two in order to perform FFT. (b) Change the value of x and n, then try your program again.

7 3. System s roots Given characteristic equation of a system ) s 3 8s 7s 0 0 ) s 3 s 7s 8 0 Please edit two programs to solve roots of the characteristic equation and record the results. (a) Using r=roots(n) to get the roots of characteristic equation. The n is characteristic equation s coefficient. (b) Please analye the two equations roots. What is their difference? (c) Is the system stable? Why? 4. Sinusoidal solution of LTI system Given a system described by the second-order differential equation: Please (a) (b) (c) d y( dy( 7 0y( 6x( dt d( Analye the system s transfer function H(s). When x(=6cos(3t+50), Edit MATLAB program to get expression of H(j3). When the system is excited by x(=6cos(3t+50), edit MATLAB program to calculate the expression of the system s ero state response y s (. Please complete the following program. n=[ ]; d=[ ]; h=polyval(n, )/polyval(d, ); ymag=6*abs(h); yphase= ; ymag yphase 5. The impulse Response of a LTI system is h( 0 t

8 Please ) Analye the Laplace transform of h( directly. ) If input function is u(, use the h( you have got from (a) to analye Y(s) directly. 3) Edit MATLAB program to calculate y( from the Y(s) you have got from (b). (a) Use syms or sym to create the symbolic variable. (b) Use ilaplace to get ILT results. 6. Analog system s frequency response An analog system s transfer function is 0.5s 0.6s H ( s). Please edit a s 0.4s MATLAB program to plot its frequency response. (a) Use w=logspace(-,) to generate a logarithmically spaced vector. (b) You can call freqs without output arguments to plot both magnitude versus frequency and phase versus frequency.

9 INFE 50 SIGNALS AND SYSTEMS Assignment 3: Discrete-time LTI System and Filter Design Name, Class&Student ID Aim. To review Discrete Signal s ZT, IZT, h(n), s(n), H( e j ), etc.. To get discrete system s eros and poles, analye the system s stability. 3. To study difference equation number solution and program s solution. 4. To design digital Butterworth filter and chebyshev filter. Introduction ) Transfer function: The transfer function is a basic -domain representation of a digital filter, expressing the filter as a ratio of two polynomials. It is the principal discrete-time model for MATLAB Signal Processing Toolbox. The transfer function model description for the -transform of a digital filter s difference equation is b() b() Y ) a() a() ( b( nb ) a( na ) nb na X ( ) Here, the constants b(i) and a(i) are the filter coefficients, and the order of the filter is the maximum of na and nb. In MATLAB, you store these coefficients in two vectors (row vectors by convention), one row vector for the numerator and one for the denominator. ) Zero-Pole Analysis: The plane function plots poles and eros of a linear system. For a system in ero-pole form, supply column vector arguments and p to plane: plane(,p). For a system in transfer function form, supply row vectors b and a as arguments to plane: plane(b,a). Note that if b and a are both scalars they will be interpreted as and p. 3) Zero-Pole-Gain: The factored or ero-pole-gain form of a transfer function is H( ) q( ) p( ) ( q())( q()) ( q( n)) k ( p())( p()) ( p( n)) By convention, MATLAB stores polynomial coefficients in row vectors and polynomial roots in column vectors. In ero-pole-gain form, therefore, the ero and pole locations for the numerator and denominator of a transfer function reside in column vectors. The factored transfer function gain k is a MATLAB scalar. The poly and roots functions convert between polynomial and ero-pole-gain representations. For example, a simple IIR filter is b=[,3,4]; a=[ 3 3 ];

10 The eros and poles of this filter are q=roots(b) q= i i p=roots(a) p= i i k=b()/a() k= Returning to the original polynomials, bb=k*poly(q) bb= aa=poly(p) aa= Note that b and a in this case represent the transfer function 3 4 H ) ( 3 For b=[ 3 4], the roots function misses the ero for equal to 0. In fact, it misses poles and eros for equal to 0 whenever the input transfer function has more poles than eros, or vice versa. This is acceptable in most cases. To circumvent the problem, however, simply append eros to make the vectors the same length before using the roots function, for example, b=[b 0]. 4) Partial Fraction Expansion (Residue Form): Each transfer function also has a corresponding partial fraction expansion or residue form representation, given by b a( ) ( ) r() r( n) ( mn) k() k() k( m n ) p() p( n) provided H() has no repeated poles. Here, n is the degree of the denominator polynomial of the rational transfer function b()/a(). If r is a pole of multiplicity s r, then H() has terms of the form r( j) p( j) r( j ) ( p( j) r( j s ) ( p( j) ) r ) ) The residue function in the Signal Processing Toolbox converts transfer functions to and from the partial fraction expansion form. The on the end of residue stands for -domain, or discrete domain. residue returns the poles in a column vector p, the residues corresponding to the poles in a column vector r, and any improper part of the original transfer function in a row vector k. resudue determines that two poles are the same if the magnitude of their difference is smaller than 0. percent of either of the poles magnitudes. Partial fraction expansion arises in signal processing as one method of finding the ) s r

11 inverse -transform of a transfer function. For example, the partial fraction 4 8 expansion of H ( ) is 6 8 b=[-4 8]; a=[ 6 8]; [r, p, k]=residue(b,a) r=- 8 p=-4 - k=[] 8 corresponding to H ( ). 4 5) Repeat Statement: The general format is for varivble=expression statement statement end The columns of the expression are stored one at a time in the variable while the following statements, up to the end, are executed. In practice, the expression is almost always of the form scalar : scalar, in which case its columns are simply scalars. The scope of the for statement is always terminated with a matching end. Please use help for to see an example. 6) Frequency Response (Digital Domain): freq uses an FFT-based algorithm to calculate the -transform frequency response of a digital filter. Specifically, the statement [h,w]=freq(b,a,n) returns the n-point complex frequency response, H ( e j ), of the digital filter. H b() b() e a() a() e b( nb ) e a( na ) e j j ( nb) ( j e ) j j ( na ) In its simplest form, freq accepts the filter coefficient vectors b and a, and an integer n to specifying the number of points at which to calculate the frequency response. freq returns the complex frequency response in vector h, and the actual frequency points in vector w in radians/second. If you call freq without output arguments, it automatically plots both magnitude versus frequency and phase versus frequency. 7) Symbolic variables: x = sym('x') creates the symbolic variable with name 'x' and stores the result in x. Statements like pi = sym('pi') and delta = sym('/0') create symbolic numbers which avoid the floating point approximations inherent in the values of pi and /0. The pi created in this way temporarily replaces the built-in numeric function with the same name. syms is short-cut for constructing symbolic objects. For example, syms arg arg... is short-hand notation for

12 arg = sym('arg'); arg = sym('arg');... 8) Filter Design: The Signal Processing Toolbox provides functions that support a range of filter design methodologies, including Infinite Impulse Response (IIR) and Finite Impulse Response (FIR) filter design problems. The toolbox provides five different types of classical IIR filters, each optimal in some way. In this assignment, you are required to try two types of them: Butterworth Filter and Chebyshev Filter. Requirements. Inverse Z Transform (IZT) Given X ) ( 0.9 ) ( 0.9 ( partial fraction expansion., please edit a MATLAB program to get the ) (c) Please use the command poly and residue. If you need help, turn to help or the introduction part on page and page 3 of this report. (d) In your report, write down the expression of X() you get by using poly. (e) In your report, write down the expression of X() you get by using poly and residue. Then, analye the expression of x[n] from X().. Stability of a discrete system The transfer function of an LTI system is H ( ). Please edit MATLAB program to calculate the characteristic-equation roots. (a) Use roots in your program. (b) Is the system stable or unstable? Why? 3. Check the solution of LTI discrete system Given y(0)=0, from difference equation y(n)-0.6y(n-)=4u(n), we can get the solution n is y( n) 0[ 0.6 ] u( n). Please () Calculate y(n) n=,,3,4 from the solution directly,. Write them down on your report. () edit MATLAB program to calculate y(n), and check them with the y(n) you have got from (a). (a) Please use for to realie the repeat statement in your program. (b) When using for, can you set y(0)=0? Why?

13 4. Digital system s frequency response A digital system s transfer function is H ( ) 0.4. Please edit a MATLAB program to plot its frequency response. Use freq(b,a,n) to get digital system s complex frequency response. The frequency response is evaluated at n points equally spaced around the upper half of the unit circle. If n isn't specified, it defaults to 5. Please set n=8 in your program. 5. Digital system s analysis Given a LTI system: y(n)=0.8y(n-)+x(n)-x(n-), please () Analye the expression of transfer function H() from the given difference equation of this system. () Edit MATLAB program to calculate its impulse response and step response. j (3) Edit MATLAB program to plot this system s frequency response H ( e ). (4) Please plot both the impulse response and step response in same figure. (a) Use dimpulse to get the impulse response of discrete-time linear systems; Use dstep to get the step response of discrete-time linear systems. (b) You can call freq without output arguments to plot both magnitude versus frequency and phase versus frequency. 6. Butterworth digital filter design Please design a 9 degree lowpass Butterworth digital filter. Its cutoff frequency is 300 H, and the sampling frequency is 000 H. (d) Use butter(n, 300/500) to design the filter. The second input argument to butter indicate the cutoff frequency, normalied to half of the sampling frequency (the Nyquist frequency). (e) Use freq(b,a,8,000) to plot the filter s frequency response. Please explain each argument s meaning in your report. 7. Lowpass Chebyshev type filter design The Chebyshev type filter minimies the absolute difference between the ideal and actual frequency response over the entire passband by incorporating an equal ripple of Rp db in the passband. Stopband response is maximally flat. The transition from passband to stopband is more rapid than for the Butterworth filter. Please design a 9 degree lowpass Chebyshev type filter. Its cutoff frequency is 300 H. Its ripple coefficient Rp=0.5. The sampling frequency is 000 H. a) Please use the command [b,a]=cheby(n,rp,wn) to design a Chebyshev type

14 filter. The meaning of the argument Wn is similar to that of Butterworth filter design. b) Use freq(b,a,5,000) to plot the filter s frequency response. 8. (Optional)Bandpass Chebyshev type filter design Please design a 0 degree bandpass Chebyshev type filter. Its ripple coefficient Rp=0.5. Its pass band is 00H<w<00H. The sampling frequency is 000 H. Please plot the filter s impulse response. (a) Please use the command [b,a]=cheby(n,rp,wn) to design a Chebyshev type filter. To design a bandpass filter, Wn should be a two-element vector: Wn = [W W]. cheby returns an order n bandpass filter with passband W < W < W. Please note that Wn must be 0.0 < Wn <.0, with.0 corresponding to half the sampling frequency. (b) Use imp(b,a,m) to computes m samples of the impulse response. Set the value of m by yourself.

15 INFE 50 SIGNALS AND SYSTEMS Assignment 4: Continuous-time LTI system Name, Class&Student ID Aim. To study time-domain analysis of continuous-time LTI system.. To get the frequency response of a continuous-time system. 3. To get the continuous-time system s eros and poles, analye the system s stability. Introduction ) Transfer function model: A Continuous-time LTI system can be described by a linear constant coefficient differential equation of the form a( N) y ( N) ( a( N -) y ( N) ( a(0) y( b( M) x The system function H (s) can easily be computed b( M ) s H s) a( N) s M b( M -) s a( N -) s ( M ) M ( N N ( b( M -) x b(0) a(0) ( M ) ( b(0) x( In MATLAB, we store the constants b (i) and a (i) in two row vectors, one for the numerator and one for the denominator. Then the tf function can be used to create transfer function model. SYS = tf(num,den) creates a continuous-time transfer function SYS with numerator NUM and denominator DEN. ) Time response: The lsim function simulates the time response of continuous linear systems to arbitrary inputs. 3) Impulse response: The impulse function plots the impulse response of a continuous-time system. 4) Step response: The step function plots the step response of a continuous-time system. 5) Frequency response: The freqs function returns the frequency response of a continuous-time system. 6) Zero-Pole analysis: The pmap function creates a pole-ero plot of a continuous-time system. 7) Please use help to find out the description of the functions above.

16 Requirements. There are four analogy filters described by the following differential equations. Please edit MATLAB program to plot the impulse response and the step response for each system. () y ( y( y( x( () y ( y( y( x ( (3) y ( y( y( x( (4) y ( y( y( x ( x(. Given a system described by the second order differential equation y ( y( 6y( x( () Plot the system s impulse response and step response using MATLAB. t () When the input signal is x( e u(, please calculate the ero-state response. 3. The system function of a third-order butterworth low-pass filter is given below H ( s) 3 s s s Edit MATLAB program to plot this system s frequency response. 4. Two causal continuous-time system s system function are described as () H ( s) 3 s s s s () H ( s) s s 3s 3s 3s Create the pole-ero plot of the system, and determine whether it s stable or not. 5. There are six continuous-time systems, and the poles of each system functions are given below. Here we assume that no ero for every system. () p 0 () p (3) p (4) p j, p j (5) p 4 j, p 4 j (6) p 4 j, p 4 j Create the pole-ero plots of the six systems above using MATLAB respectively. And plot the impulse-response for every system. Based on the plots, analye how the pole-ero s position impacts system s impulse-response characteristics.

17 6. We got three continuous-time systems, and their system functions are described as, () H ( s) s s 7 s 8 () H ( s) s s 7 s 8 (3) H ( s) s s 7 The systems above have the same poles, but different eros. Plot each system s pole-ero plots and impulse-response respectively using MATLAB. Based on the plots, analye how the pole-ero s position impacts system s impulse-response characteristics.

18 INFE 50 SIGNALS AND SYSTEMS Assignment 5: Discrete-time LTI system Name, Class&Student ID Aim. To study time-domain analysis of discrete-time LTI system.. To get the frequency response of a discrete-time system. 3. To get the discrete-time system s eros and poles, analye the system s stability. 4. To compute the convolution between two finite-duration sequence. Introduction ) Transfer function model: A discrete-time LTI system can be described by a linear constant coefficient difference equation of the form a( 0) a() y( n ) a( N) y( n N) b(0) b() x( n ) b( M) x( n M) The system function H (s) can easily be computed b(0) b() H s) a(0) a() ( b( M ) a( N) M N In MATLAB, we store the constants b (i) and a (i) in two row vectors, one for the numerator and one for the denominator. ) Time response: The filter function compute the output signal of discrete-time systems to an input signal and is invoked by y = filter(b,a,x) Here, vector X is the input signal, and the vector b and a is respectively numerator coefficient and denominator coefficient. 3) Impulse response: The imp function plots the impulse response of a discrete -time system and is invoked by imp(b,a). 4) Convolution: The conv function compute the convolution between two finite-duration sequence. 5) Frequency response: The freq function returns the frequency response of a discrete-time system.

19 6) Zero-Pole analysis: The pmap function plots poles and eros of a discrete-time system. 7) Please use help to find out the description of the functions above. Requirements. Given the difference equations of discrete-time systems as follows. Plot the system s impulse response respectively. () y( n) 3y( n ) y( n ) x( n) () y ( n) 0.5y( n ) 0.8y( n ) x( n) 3x( n ). Given the difference equations of the system as follow, y( n) y( n ) 0.5y( n ) x( n) Please Plot the impulse response and the step response of this system using MATLAB. 3. Please edit MATLAB program to calculate the convolution between the following two sequences. x ( n) {,,,} - n x ( n) 0 其他 4. Given a discrete-time system as follow y( n).004y( n ).464y( n ) y( n 3) 0.044x( n ) 0.037x( n ) Plot the system s frequency response using MATLAB. 5. Given the system functions of the causal discrete-time systems as follows, () H ( ) () H ) ( 4 Create the Zero-Pole plots for each system and determine if the system is stable. 6. Given the eros and poles of six discrete-time systems as follows, () 0, p 0. 5 () 0, p (3) 0, p. 5

20 (4) 0 (5) 0 6 6, p 0. e, p 0. e 8 j 8 8 8, p e, p e j 3 j j j 4 4 (6) 0, p. e, p. e 3 j Create the pole-ero plots of the six systems above using MATLAB respectively. And plot the impulse-response for every system. Based on the plots, analye how the pole-ero s position impacts system s impulse-response characteristics. 7. Given the system functions of the discrete-time systems as follows, () H( ) () H( ) ( 0.8e ( 0.8e ( ) j 6 )( 0.8e ( ) j 6 )( 0.8e j 6 j 6 ) ) The systems above have the same poles, but different eros. Plot each system s pole-ero plots and impulse-response respectively using MATLAB. Based on the plots, analye how the position of eros of the system influence the characteristic of the impulse-response.