Exercises in Signals

1 Exercises in Signals , Systems, and TransformsIvan W. SelesnickLast edit: October 27, 2014 Contents1 discrete - time Signals and System More Z Inverse Di erence Complex Frequency Summary Simple System More Continuous- time Signals and System Laplace Di erential Complex Frequency Simple System fourier fourier fourier The Sampling Theorem15011 discrete - time Signals and an accurate sketch of each of the discrete - time Signals (a)x(n)=u(n+ 3) + (n 1)(b)x(n)= (n+ 3) + (n 1)(c)x(n)=2n (n 4)(d)x(n)=2n u( n 2)(e)x(n)=( 1)nu( n 4).(f)x(n)=2 (n+ 4) (n 2) +u(n 3)(g)x(n)=1Xk=04 (n 3k 1)(h)x(n)=1Xk= 1( 1)k (n 3k) a sketch of each of the following Signals (a)x(n)=1Xk= 1( )|k| (n k)(b)x(n) = cos( n)u(n)(c)x(n)=u(n) 2u(n 4) +u(n 8) (n),x1(n),x2(n), andx3(n)wherex(n)=u(n+ 4) u(n),x1(n)=x(n 3),x2(n)=x(5 n),x3(n)=nXk= 1x(k) (n) andx1(n)wherex(n)=( )nu(n),x1(n)=nXk= 1x(k) (n) andx1(n)wherex(n)=n[ (n 5) + (n 3)],x1(n)=nXk= 1x(k) a sketch of each of the following Signals (a)f(n)=1Xk=0( )k (n 3k)(b)g(n)=1Xk= 1( )|k| (n 3k)(c)x(n) = cos( n)u(n)(d)x(n) = cos( n)u(n) discrete - time Signals in plot the discrete - time impulse function:n = -10:10;f = (n == 0);stem(n,f)Usestemto plot the discrete - time step function:f = (n >= 0);stem(n,f)Make stem plots of the following Signals .

2 Decide for yourself what the range ofnshould (n)=u(n) u(n 4)(1)g(n)=r(n) 2r(n 5) +r(n 10) wherer(n):=nu(n)(2)x(n)= (n) 2 (n 4)(3)y(n)=( )n(u(n) u(n 20))(4)v(n) = cos( n)u(n)(5) System discrete - time system may be classified as follows: memoryless/with memory causal/noncausal linear/nonlinear time -invariant/ time -varying BIBO stable/unstableClassify each of the following discrete -times systems.(a)y(n) = cos(x(n)).(b)y(n)=2n2x(n)+nx(n+ 1).(c)y(n) = max{x(n),x(n+ 1)}Note: the notation max{a,b}means for example; max{4,6}= 6.(d)y(n)= x(n)whennis evenx(n 1) whennis odd(e)y(n)=x(n)+2x(n 1) 3x(n 2).(f)y(n)=1Xk=0(1/2)kx(n k).That is,y(n)=x(n)+(1/2)x(n 1) + (1/4)x(n 2) + (g)y(n)=x(2n) discrete - time system is described by the following ruley(n)= (2n)+ (2n 1)wherexis the input signal, andythe output signal.

3 (a)Sketch the output signal,y(n), produced by the 4-point input signal,x(n) illustrated (n)5(b)Sketch the output signal,y(n), produced by the 4-point input signal,x(n) illustrated (n)(c)Classify the system discrete - time system is described by the following ruley(n)=(x(n),whennis an even integer x(n),whennis an odd integerwherexis the input signal, andythe output signal.(a)Sketch the output signal,y(n), produced by the 5-point input signal,x(n) illustrated (n)(b)Classify the system discrete - time system is described by the following ruley(n)=( 1)nx(n)+2x(n 1)wherexis the input signal, andythe output signal.(a)Accurately sketch the output signal,y(n), produced by the input signalx(n) illustrated (n)(b)Classify the system the output of an LTI impulse response of a discrete - time LTI system ish(n)=2 (n)+3 (n 1) + (n 2).)

4 Find and sketch the output of this system when the input is the signalx(n)= (n)+3 (n 1) + 2 (n 2). a discrete - time LTI system described by the ruley(n)=x(n 5) +12x(n 7).What is the impulse responseh(n) of this system? impulse response of a discrete - time LTI system ish(n)= (n)+2 (n 1) + (n 2).Sketch the output of this system when the input isx(n)=1Xk=0 (n 4k). impulse response of a discrete - time LTI system ish(n)=2 (n) (n 4).Find and sketch the output of this system when the input is the step functionx(n)=u(n). the discrete - time LTI system with impulse responseh(n)=nu(n).(a)Find and sketch the outputy(n)whentheinputx(n)isx(n)= (n) 2 (n 5) + (n 10).(b)Classify the system as BIBO the output of an LTI impulse responseh(n) of an LTI system is given byh(n)= 23 nu(n).Find and sketch the outputy(n) when the input is given by(a)x(n)= (n)(b)x(n)= (n 2) the LTI system with impulse responseh(t) = cos ( t)u(n),find and sketch the step responses(t) and classify the system as BIBO the LTI system with impulse responseh(n)= (n 1).

5 (a)Find and sketch the outputy(n)whentheinputx(n) is the impulse train with period 6,x(n)=1Xk= 1 (n 6k).(b)Classify the system as BIBO LTI system is described by the following equationy(n)=1Xk=0 13 kx(n k).Sketch the impulse responseh(n) of this the parallel combination of two LTI (n)-h1(n)x(n)?6l+-y(n)You are told thath1(n)=u(n) 2u(n 1) +u(n 2).You observe that the step response of the total system iss(n)=2r(n) 3r(n 1) +r(n 2)wherer(n)=nu(n). Find and sketchh2(n). impulse response of a discrete - time LTI system is given byh(n)= 1ifnis a positive prime number0 otherwise (a)Is the system causal?(b)Is the system BIBO stable? observe an unknown LTI system and notice thatu(n) u(n 2)-S- (n 1) 14 (n 4)Sketch the step responses(n). The step response is the system output when the input is the step functionu(n).

6 An LTI system it is known that input signalx(n)= (n)+3 (n 1)produces the following output signal:y(n)= 12 nu(n).What is the output signal when the following input signal is applied to the system?x2(n)=2 (n 2) + 6 (n 3) More and sketch the convolutionx(n)=(f g)(n)where(a)f(n)=2 (n+ 10) + 2 (n 10)g(n)=3 (n+ 5) + 3 (n 5)10(b)f(n)= (n 4) (n 1)g(n)=2 (n 4) (n 1)(c)f(n)= (n+ 2) (n+ 1) (n)g(n)= (n)+ (n+ 1) + (n+ 2)(d)f(n)=4g(n)= (n)+2 (n 1) + (n 2).(e)f(n)= (n)+ (n 1) + 2 (n 2)g(n)= (n 2) (n 3).(f)f(n)=( 1)ng(n)= (n)+ (n 1). impulse response of a discrete - time LTI system ish(n)=u(n) u(n 5).Sketch the output of this system when the input isx(n)=1Xk=0 (n 5k). signalfis given byf(n) = cos 2n .The signalgis (n)Sketch the signal,x(n), obtained by convolvingf(n) andg(n),x(n)=(f g)(n).

7 Signalsfandgare given byf(n)=2,g(n)= 12 nu(n).Sketch the signal,x(n), obtained by convolvingf(n) andg(n),x(n)=(f g)(n). signalsf(n) andg(n) are shown: 4 3 2 101234012312321f(n)n 4 3 2 101234 2 10123 12 1g(n)nSketch the convolutionx(n)=f(n) g(n). the convolution of the discrete - time signalx(n)2321-2-10123456nx(n)with each of the following Signals .(a)f(n)=2 (n) (n 1)(b)f(n)=u(n)(c)f(n)= (d)f(n)=1Xk= 1 (n 5k) signalsfandgare defined as:f(n)=anu(n)g(n)=f( n)=a nu( n)Find the convolution:x(n)=(f g)(n)Plotf,g, andxwhena= You may use a computer for average filterhas the impulse responseh(n)= 1/N0 n N 10otherwiseUse the Matlabconvcommand to computey(n)=h(n) h(n)forN=5,10,20, and in each case make a stem plot ofh(n) andy(n).What is the general expression fory(n)? convolution of two finite length Signals can be written as a matrix vector product.

8 Look at the documentationfor the Matlabconvmtxcommand and the following Matlab code that shows the convolution of two Signals by(1) a matrix vector product and (2) theconvcommand. Describe the form of the convolution matrix and whyit works.>> x = [1 4 2 5]; h = [1 3 -1 2];>> convmtx(h ,4)*x ans =1713921-110>> conv(h,x) ans = convolutiony=h g,wherehandgare finite-length Signals , can be represented as a matrix-vector product,y=HgwhereHis a convolution matrix. In MATLAB, a convolution matrixHcan be obtained with thecommandconvmtx(h(:), K).Given finite-length sequenceshandx,defineH = convmtx(h(:), M)whereMis such that the matrix-vector productHTxis defined, whereHTdenotes the transpose terms of convolution, what does the matrix-vector productHTxrepresent?Write a MATLAB function to computeHTxusing the functionconvand without creating the to your function should be vectors, (n)=u(n) u(n 5)g(n)=r(n) 2r(n 5) +r(n 10).

9 Wherer(n):=nu(n).In MATLAB, use theconvfunction to compute the following convolutions. Use thestemfunction to plot theresults. Be aware about the lengths of the Signals . Make sure the horizontal axes in your plots are correct.(a)f(n) f(n)(b)f(n) f(n) f(n)(c)f(n) g(n)(d)g(n) (n)(e)g(n) g(n)Comment on your observations: Do you see any relationship betweenf(n) f(n) andg(n) ? Comparef(n)withf(n) f(n) and withf(n) f(n) f(n). What happens as you repeatedly convolve this signal with itself?Use the commandstitle,xlabel,ylabelto label the axes of your of non-causal Signals in MATLABNote that both of these Signals start to the left ofn= (n)=3 (n+ 2) (n 1) + 2 (n 3)(6)g(n)=u(n+ 4) u(n 3)(7)First, plot the signalsf,g, andf gby hand, without using MATLAB. Note the start and end , use MATLAB to make plots off,g, andf g.

10 Be aware that theconvfunction increases the length turn in: The plots off(n),g(n),x(n), and your Matlab commands to create the data byN-point the data the course website. Load the data into Matlab using the see your variables. One of the variables will beDataEOG. For convenience, renameit toxby typing:x = DataEOG;This signal comes from measuring electrical Signals from the brain of a a stem plot of the signalx(n). You will see it doesn t look good because there are so many points. Makea plot ofx(n)usingtheplotcommand. As you can see, for long Signals we get a better plot using theplotcommand. Although discrete - time Signals are most appropriately displayed with thestemcommand, forlongdiscrete- time Signals (like this one) we use the plot command for better a simple impulse response for an LTI system:h = ones(1,11)/11;Compute the convolution ofhandx:y = conv(x, h);Make a MATLAB plot of the outputy.