EEO 401 Digital Signal Prcessing Prf. Mark Fwler Intrductin Nte Set #1 ading Assignment: Ch. 1 f Prakis & Manlakis 1/13
Mdern systems generally DSP Scenari get a cntinuus-time signal frm a sensr a cnt.-time system mdifies the signal an analg-t-digital cnverter (ADC r A-t-D) sample the signal t create a discrete-time signal a stream f numbers A discrete-time system t d the prcessing and then (if desired) cnvert back t analg (nt shwn here) x(t) x[n] y[n] y(t) Sensr Analg Electrnics t ADC n DSP n DAC t Physical C-T Signal Electrical C-T Signal Electrical C-T System Electrical C-T Signal Electrical D-T Signal Electrical D-T System Electrical D-T Signal C-T Signal 2/13
Our fcus will be n Sampling thery Frequency-Dmain Mdels fr DT Signals Frequency-Dmain Mdels fr DT Systems Prcessing structures fr implementing DSP systems Methds fr designing DSP systems Ensures that samples are equivalent t CT signal Prvides math t understand HOW the DSP wrks Sectin 1.2 Classificatin f Signals Multichannel vs. Single Channel CT vs DT Discrete-Valued vs Cntinuus-Valued Randm vs Deterministic Gives practical ways t MAKE DSP wrk 3/13
Transfrms & Ntatin Furier Transfrm fr CT Signals F j2π Ft X ( F) x() t e dt = F j2π Ft x t X F e df () = ( ) Prakis & Manlakis dn t use this superscript Ntatin. I brrwed it frm Prat s DSP Bk Furier Transfrm fr DT Signals X f ( ω) = xne [ ] n= jωn π 1 f jωn xn [ ] X ( ) e d = 2π π ω Discrete Time Furier Transfrm θ Set z = e jω Z Transfrm fr DT Signals X z ( z) = xnz [ ] n= n Inverse ZT dne using partial fractins & a ZT table Discrete Furier Transfrm fr DT Signals N 1 d j2 π kn/ N X [ k] = xne [ ] k = 0,1,2,..., N 1 n= 0 1 xn X ke n N N N 1 d j2 π kn/ N [ ] = [ ] = 0,1,2,..., 1 k= 0 Discrete Furier Transfrm 4/13
Discrete-Time System latinships Time Dmain Z / Freq Dmain Inspect Blck Diagram Inspect Ple/Zer Diagram Rts Difference Equatin ZT (Thery) Inspect (Practice) Transfer Functin H z ( z) = b 0 + b 1+ a 1 1z 1 1z + + b + + a q qz p pz p yn [ ] = ayn [ i] + bxn [ i] i i= 1 i= 0 q i ZT Unit Circle z = e jω pulse spnse h[n] DTFT Frequency spnse H f ( ) = H z ( z) jω ω z = e 5/13
Sinusidal Time Functin A sinusid is cmpletely defined by its three parameters: Amplitude A (fr us typically in vlts r amps but culd be ther unit) Frequency F in Hz Phase φ in rad (nt degrees!) x( t) = Asin(2 πft+ φ) (Similar fr csine) (rad/cycle) (cycle/sec) sec + rad = rad T is the Perid in secnds (actually secnds/cycle) f the sinusid t is a time shift it is related the phase φ T = 1/ F 6/13
Cmplex Sinusidal Time Functin In many cases it is desirable t write a real-valued sinusid in terms f cmplex-valued sinusids. This is a math trick that believe it r nt! makes things easier t wrk with!!! A x( t) = Acs(2 πft + φ) = e + e 2 j(2 πft+ φ) j(2 πft+ φ) This cmes frm Euler s Frmula: j e θ cs( θ ) = e jθ + e 2 jθ 2cs( θ ) Anther frm f Euler s Frmula: e e jθ jθ = cs( θ) + jsin( θ) j e θ = cs( θ) jsin( θ) jsin(θ) -jsin(θ) j e θ cs( θ ) j e θ 7/13
Explring the Cmplex Sinusidal Terms A x( t) = Acs(2 πft + φ) = e + e 2 aginary part always cancels! j(2 πft+ φ) e j(2 πft+ φ) j(2 πft+ φ) Tw cmplex values with ppsite angles Psitive Frequency Term Rtate ppsite directins due t negative sign e j(2 π Ft+ φ ) j( 2 π Ft φ ) = e Negative Frequency Term Here is a link t a Quicktime mvie f these rtating http://www.cic.unb.br/~mylene/psmm/dspfirst/chapters/2sines/dems/phasrs/graphics/phasrsn.mv Link t anther Web Dem f this 1. Open the web page 2. Click n the bx at the tp labeled Tw 8/13
Sampling Sinusids DT Sinusids Sensr Analg Electrnics x(t) t ADC x[n] n Digital Elec. (Cmputer) x[n] is just a stream f numbers Optinal DAC ADC Clck sets hw ften samples are taken Hw clsely shuld the samples be spaced?? At first thught we might think we need t have the samples still lk like the riginal sinusid But that turns ut t be excessive, as ur thery will shw eventually shw. Lking at the samples x[n] abve they dn t quite really lk like a sinusid yet they are taken at a rate suitable fr mst applicatins! S hw d we determine hw fast we need t sample??? 9/13
DT Samples. What CT Sinusid did they cme frm???? t (sec) They culd have cme frm this blue ne t (sec) But They culd have cme frm this RED ne!!! t (sec) Thus if we want t be able t tell these tw apart we need t sample faster!! 10/13
Let T s be the time spacing between samples Then F s = 1/T s as the sampling frequency in samples/sec. Then if we have a CT sinusid x(t) = cs(2πf t) that is sampled we have = x[ n] = x( nts) = cs(2 π FT sn) x( t) cs(2 π Ft) Discrete-Time Sinusid F xn [ ] = cs( ωn) ω 2π F Units are rad/sample s ω S t help visualize this: 1 cs( n ) e ω e ω ω = 2 + j n j n j n 8 e π π 8 j n 8 e π 11/13
n = 1 n = 2 n = 3 n = 4 ω = ω = 7π 8 9π 8 S a DT frequency > π rad/sample lks exactly like sme ther frequency < π. This is called Aliasing. n = 1 n = 2 n = 3 n = 4 12/13
S t avid this aliasing when sampling a CT sinusid t make a DT sinusid we must require that: F F s s 2 F < ω < 2π = π 2 Thus fr prper sampling we need t chse ur sampling rate t be mre than duble the highest frequency we expect!!! Aside: This is cnsistent with sme real-wrld facts yu may knw abut: High-Fidelity Audi cntains frequencies up t nly abut 20 khz CD digital audi has a sampling frequency f F s = 44.k Hz > 2x20kHz F s cycle/sample rad/sample cycle/secnd Frm textbk by Prakis & Manlakis 13/13