EECE 301 Signals & Systems Prof. Mark Fowler1/9EECE 301 Signals & SystemsProf. Mark FowlerNote Set 11aExample: Using D-T Convolution to Simulate the Effect of an RC Circuit on a Guitar Signal2/9Ch. 1 IntroC-T Signal ModelFunctions on Real LineD-T Signal ModelFunctions on IntegersSystem PropertiesLTICausalEtcCh. 2 Diff EqsC-T System ModelDifferential EquationsD-T Signal ModelDifference EquationsZero-State ResponseZero-Input ResponseCharacteristic Eq.Ch. 2 ConvolutionC-T System ModelConvolution IntegralD-T System ModelConvolution SumCh. 3: CT Fourier SignalModelsFourier SeriesPeriodic SignalsFourier Transform (CTFT)Non-Periodic SignalsNew System ModelNew SignalModelsCh. 5: CT Fourier SystemModelsFrequency ResponseBased on Fourier TransformNew System ModelCh. 4: DT Fourier SignalModelsDTFT(for “Hand” Analysis)DFT & FFT(for Computer Analysis)New SignalModelPowerful Analysis ToolCh. 6 & 8: Laplace Models for CTSignals & SystemsTransfer FunctionNew System ModelCh. 7: Z Trans.Models for DTSignals & SystemsTransfer FunctionNew SystemModelCh. 5: DT Fourier System ModelsFreq. Response for DTBased on DTFTNew System ModelCourse Flow DiagramThe arrows here show conceptual flow between ideas. Note the parallel structure between the pink blocks (C-T Freq. Analysis) and the blue blocks (D-T Freq. Analysis).3/9The m-file RC_filt.m is available for download from the course website.The wav file guitar1.wav is also available there.To run it all you need to do is: 1. Download these files to a folder on your computera. Make sure Matlab’s current directory is the folder where you have put the files (change it if needed)2. At the Matlab command line type: RC_filter3. The m-file will runa. Follow the prompts given in the Matlab Command Window to play the original signal and the signal after going through the RC Circuitb. Finally, the m-file will plot the impulse response h(t) for the RC CircuitExample: D-T Processing to Simulate an RC Filter4/9Guitar AmpPhysical Set-Up of Guitar and RC CircuitVpickupR1Guitar’s Vol. ControlR2CGuitar’sTone ControlRinUse TheveninEquivalent CircuitAssume Amp’s Input Impedence is Very Large (i.e., Open Circuit)Guitar AmpVthRthR2C5/9Circuit Model for Physical Set-UpVthR = Rth+ R2COutput SignalInput SignalWe know that this RC circuit has an impulse response given by:RCtRCeth/1)(−=R = 100 kΩC = 5 nF6/9Modeling the C-T Circuit Using an Approximate D-T SystemWe can use D-T convolution to numerically approximate a C-T convolution.Recall…• C-T Convolution involves an Integral • D-T Convolution involves a Summationf(x)x∫∑=ΔΔ≈biifdxxf050)()(0b = 6ΔΔ 2Δ 3Δ 4Δ 5ΔΔObviously… to be reasonably accurate we need Δ quite smallRecall from Calc I…Integration was defined as the limit of a series of SummationsSo… to approximately compute the integral we only need Samplesof the function!7/9So… for our case we have:RCnTRCTenh/][−=This result says that if • h(t) is the C-T system’s impulse response • T is the spacing between time samples in the D-T simulationSo apply this idea to C-T convolution:TiTthiTxdthxtyi∑∫∞−∞=∞∞−−≈−=)()()()()(τττNow… we’d be happy to get closely spaced samples of the output… so replace t by nT()][*][][][)()()( nThnxTinhixTiTnThiTxnTyii=−=−≈∑∑∞−∞=∞−∞=D-T Convolution!!)(][ nTThnh=Then the equivalent D-T system has an impulse response given by:8/9Simulating the Circuit in MatlabC=5e-9;R=100e3; %%% sets RC = 0.5 msRC=R*C; %% compute RC time constant for use later[x,Fs]=wavread('guitar1.wav'); % read in wave file of guitar recordingdel_t=1/Fs; % Fs is sampling rate read in by wavread.m so invert it to get time spacingt=(0:500)*del_t; %% create 501 point time sample vectorh=(1/RC)*exp(-t/RC); %% compute impulse response of this RC circuit at the 501 time points%% use del_t*h(nTs) to make D-T conv approximate C-T conv%% See Eq. (3.64) in Sect 3-5 of booky=conv(x,del_t*h); %% this D-T convolution approximates the C-T convolution that %% the RC circuit doesNow run RC_filt.m to hear the effect of this filter on the guitar signal. You should note that the guitar now sounds muffled… that is because the filter has removed some of the high frequency content.You can experiment by trying different values for R to get different RCvalues… the so-called “cut-off” frequency is given by fc= 1/2πRC (Hz)9/9Big PictureA physical RC circuit does convolution for itselfBut here we used D-T convolution to approximate C-T convolution…Then used MATLAB to numerically compute the approximating D-T convolution to numerically analyze the circuit.These kinds of numerical analyses are wide-spread in real-world design