DOC PREVIEW
BU EECE 301 - Note Set #32

This preview shows page 1-2-17-18-19-36-37 out of 37 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 37 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 37 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 37 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 37 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 37 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 37 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 37 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 37 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

EECE 301 Signals & Systems Prof. Mark FowlerDefining the Decibel“Extending” the Decibel1/37EECE 301 Signals & SystemsProf. Mark FowlerNote Set #32• C-T Systems: Transfer Function … and Frequency Response• Reading Assignment: Section 8.5 of Kamen and Heck2/37Ch. 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 Signal ModelPowerful 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).3/378.5 Frequency Response FunctionWe have seen that there are two similar tools that relate the output signal to the input signal:Transfer Function: H(s) Frequency Response: H(ω) ⎩⎨⎧∠ωωωω.)(.)(vsHvsHWe could just plot over ω> 0, because we know about symmetriesIn analysis/design of systems and circuits it helps to look at plots of:If the system is stable we know that we can use H(ω)as a tool……and we can easily get H(ω) from H(s) by replacing s → jωIt is, of course, easy to use computers to compute the data and plot it…Anyone can be trained to do that… good engineers are valuable because they understandwhat the plots show!!!!Next4/37)(tx)(ty+−Recall:)/1()/(/1)(2LCsLRsLCsH++=Example of Computing the Frequency ResponseRecall the series RLC circuit…Given specific component values: R = 20Ω L = 1mH C = 1μF942910)102(10)(+×+=sssHThe transfer function then becomes:Or… Use Matlab’s “freqs” routineNow it is possible to replace s → jωand then use general numerical S/W to compute the frequency response….Next5/37FREQS Laplace-transform (s-domain) frequency response.H = freqs(B,A,w) returns the complex frequency response vector H of the filter B/A:)(...)2()1()(...)2()1()()()(2121naasasanbbsbsbsAsBsHnananbnb++++++==−−−−given the numerator and denominator coefficients in vectors B and A. 0 5 10 15 20 25 30 3500.511.52f (kHz)|H(f)|0 5 10 15 20 25 30 35-4-3-2-10f (kHz)<H(f)>> w=0:100:20e4;>> H=freqs(1e9,[1 2e4 1e9],w);>> subplot(2,1,1)>> plot((w/(2*pi))/1e3,abs(H)) >>%%% Plots are vs. f in kHz>> subplot(2,1,2)>> plot((w/(2*pi))/1e3,angle(H))Create w in units of rad/sec – convert to kHz942910)102(10)(+×+=sssHNext6/370 5 10 15 20 25 30 3500.511.52f (kHz)|H(f)|Although the previous plots are correct, there are two problems…Suppose we are interested in using this filter in an audio application:1. We may be just as interested in 0 – 5 kHz as we are in 5 – 30 kHz─ But this plot has the 0 – 5 kHz region all “scrunched up”2. Values of |H| of, say, 0.1 and 0.01 affect the signal by significantly different amounts ─ But they show up looking virtually the same on the plot aboveNext7/3710-1100101102103104105-60-50-40-30-20-10010f (Hz)|H(f)| (dB)10-1100101102103104105-3.5-3-2.5-2-1.5-1-0.500.5f (Hz)<H(f)0 5 10 15 20 25 30 3500.511.52f (kHz)|H(f)|0 5 10 15 20 25 30 35-4-3-2-10f (kHz)<H(f)The fixes to these problems are:1. Plot on a logarithmic frequency axis…2. Plot the magnitude in dB…Our Original PlotsBode PlotsDecibelsSuch plots are “Bode Plots”… named after the engineer who introduced them0.10.01Next8/37To convert |H(ω)| into dB we use 20 log10(|H(ω)|)……we’ll see more about this in just a bitf=logspace(-1,5,2000); % create log-spaced frequencies in Hzw=2*pi*f; % convert into rad/sec for use in freqsH=freqs(1e9,[1 2e4 1e9],w); % compute H as beforesemilogx(f,20*log10(abs(H))) % plot with log x-axisw/ 20 log10( ) for dBsubplot(2,1,2)semilogx(f,angle(H))Here are the matlab commands to make these plots…2000 points from 10-1Hz to 105Hz, equally spaced on a log axis Use 20 because |H| is not a power gain(See next few pages)“semilogx” makes a plot with a log x-axis and linear (ordinary) y-axis“semilogx” – gives log x-axis, linear y-axis“semilogy” – gives linear x-axis, log y-axis“loglog” – gives log x-axis, log y-axisNext9/37A Semi-Log Axis10 1031041051060–10–20–30–40–5010203040ω (rad/sec)|H(ω)| (dB)20406080102200400600800100Next10/37• Definition: use “decibels” as a logarithmic unitof measure for a ratio between two powersDefining the Decibeldecibelbel2110log10⎟⎟⎠⎞⎜⎜⎝⎛PPNext11/37P1/P2 (non-dB)P1/P2(dB)1000 = 10330 dB100 = 10220 dB10 = 10110 dB1 = 1000 dB0.1 = 10-1–10 dB0.01 = 10-2–20 dB0.001 = 10-3–30 dBKnow These!!Powers of 10 are easy to convert to dB!!Decibel Power Rules0 dB is “unity power gain”10 dB is “changes power by 10x”20 dB is “changes power by 100x”30 dB is “changes power by 1000x”-10 dB is “changes power by 0.1x”-20 dB is “changes power by 0.01x”-30 dB is “changes power by 0.001x”P1/P2 = 2 Ä ~ 3 dBAnother “Rule” to Know!!P1/P2 = 1/2 Ä ~ -3 dBNext12/37• Even though dB is defined for power we can extend it for use with voltages and currents:– We “imagine” voltages to be compared are across the same resistance“Extending” the Decibel⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛2110222110102110log20log10log10log102221VVVVPPRVRV⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛21102221102221102110log20log10log10log10IIIIRIRIPPUse “20” for V & I, but use “10” for PTo apply dB to a “power quantity”: use 10 log10( )To apply dB to a “non-power quantity”: use 20 log10( )Next13/37V1/V2 (non-dB)V1/V2(dB)1000 = 10360 dB100 = 10240 dB10 = 10120 dB1 = 1000 dB0.1 = 10-1–20 dB0.01 = 10-2–40 dB0.001 = 10-3–60 dBKnow These!!Decibel Non-Power RulesV1/V2


View Full Document

BU EECE 301 - Note Set #32

Download Note Set #32
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Note Set #32 and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Note Set #32 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?