Signal Processing - PRE

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Convolution Revisited ignal Processing - PRE, Filtering, ampling & Quantization hannon Energy Envelope Processing Chain LR mean LR mean BP d/dn LRmean BP EE econds Copyright Cameron Rodriguez teps

1 Convolution Revisited ignal Processing - PRE, Filtering, ampling & Quantization hannon Energy Envelope Processing Chain LR mean LR mean BP d/dn LRmean BP EE econds Copyright Cameron Rodriguez teps Acquiring The ignal Filtering The ignal Ampliying a signal ampling an Analog ignal & Quantizing The ignal s(t) Aquire Filter Ampliy Digitize s*(t) s (t) s (t) s[n] Copyright Cameron Rodriguez

2 The goal is to measure the signal - s(t) without altering it. Think about measuring voltage and current in a circuit. Where and how are both important. Let s use EEG as an example, how can you measure the microvolt dierences on the scalp without disturbing system? That is, how is it possible to avoid the having the head serve as current source in the data acquisition? Aquire Recall the Instrumentation Ampliier Copyright Cameron Rodriguez Which Op-Amp Circuit Does this look like? What does this portion o the circuit do? And This One? What does this portion o the circuit do?

And this one? What does this portion o the circuit do? Beyond just measuring the signal it is also important to do so in a ashion the minimizes the noise.

3 And this one? What does this portion o the circuit do? Beyond just measuring the signal it is also important to do so in a ashion the minimizes the noise. Where possible it is better to avoid or minimize the inclusion noise in the signal Examples o doing this in EEG include: Minimizing contact impedance Minimizing contact movement with respect to the scalp Common mode rejection Driven eedback signal Cable hielding Copyright Cameron Rodriguez Filtering s(t) = Trueignal s * (t) = s(t) + Error(t) + ε N (t) = Measured ignal s * (t) = s(t) + ε (t) Copyright Cameron Rodriguez

4 ampling & Filtering - pectral Repeats ampling & Filtering - Nyquist-hannon Assumption : The sampled signal is band limited to some requency (B) (!) Assumption : The extent o the band is know ignal: s(t) ampled ignal = s(t) δ (t mtsample ) ( ample Points: δ (t mtsample ), m int, Tsample = sample period ) FT s(t)δ (t mtsample ) (ω ) δ ω Tsample = (ω Tsample ) -B B Copyright Cameron Rodriguez ampling & Filtering - Nyquist-hannon ampling & Filtering - Nyquist-hannon (!-:F m), F > B F in =., Fsample = - -B B F F -F -F -F -F -B B F F F F F -F -F -F -F -B BF F F F F Fs ec F in =., Fsample =. ec Copyright Cameron Rodriguez - -F - -F ec F in =., Fsample =. (!-:F m), F < B - -F ec F in =., Fsample =. -F - (!-:F m), F = B ec F in =., Fsample = -F Copyright Cameron Rodriguez Copyright Cameron Rodriguez

5 ampling & Filtering - pectral Repeats ampling & Filtering - Anti-Aliasing Filter ignal: s(t) ample Points: δ (t mtsample ), m int, Tsample = sample period Yah yah, thats all well and good, but what i my signal is not band limited or you don't know or need the ull bandwidth o the signal ampled ignal = s(t) δ (t mtsample ) Fsample = Tsample ( Is noise band limited? Isn't that part o the sampled signal as well? ) FT s(t)δ (t mtsample ) (ω ) δ ω Tsample = (ω Tsample ) = (ω π mfsample ) How should we handle this? Bandlimit o ignal: B in () Fsample > B Copyright Cameron Rodriguez ampling & Filtering - Anti-Aliasing Filter Copyright Cameron Rodriguez ampling & Filtering - Anti-Aliasing Filter Filter Type / hape [!]+[!] [!]. nd Order Filter Frequency Response butter cheby cheby ellip (Freq) Attenuation (db) [!]+[!] [!] log( (Freq) ) Powerin oltage in oltage in oltage in db = log = log = log = log Powerout oltage out oltage out oltage out Copyright Cameron Rodriguez Copyright Cameron Rodriguez

6 ampling & Filtering - Anti-Aliasing Filter ` Filter Type / hape Time Domain ignal th Order Filter Frequency Response AU butter cheby cheby ellip Attenuation (db) Magnitude pectrum. - (Freq) Copyright Cameron Rodriguez Filters - The Danger o Assumptions Copyright Cameron Rodriguez Z-Transorm - The Discrete Laplace (sort o) Time Domain ignal X[z] = Z { x[n]} = AU x[n]z n n= z = Ae jφ = A(cosφ + j sin φ ) Laplace Transorm (Freq) x(t) Magnitude pectrum. x[n] X(s) x(t m)u(t m) e. Z-Transorm sm X(s) X[Z ] x[n m] z m X[Z ] Copyright Cameron Rodriguez

7 Z-Transorm - Causal Filter Z-Transorm - Centered Filter y[n] = α x[n]+ α x[n ]+ α x[n ]+... Y[z] = α X[z]+ α z X[z]+ α z X[z]+... y[n] = x[n] h[n] Y[z] = X[x]H[z] H[z] = Y[z] X[z] = α + α z + α z +... z = e jω H[e jω ] = α + α cos( jω ) α j sin( jω ) + α cos( jω ) α j sin( jω ) +... H[e jω ] = H[e jω ] ( )!H[e jω ] = arctan I(H[e jω ]) R(H[e jω ]) y[n] = α x[n]+ α x[n ]+ α x[n +]+ α x[n ]+ α x[n + ]... Y[z] = α X[z]+ α z X[z]+ α z X[z]+ α z X[z]+ α z X[z]+... y[n] = x[n] h[n] Y[z] = X[x]H[z] H[z] = Y[z] X[z] = α + α z + α z + α z + α z +... z = e jω H[e jω ] = α + α cos( jω ) + α cos( jω ) +... H[e jω ] = H[e jω ] ( ) =!H[e jω ] = arctan I(H[e jω ]) R(H[e jω ]) Phase hit. Magnitude o H(z) pi Phase o H(z). pi/ H(z). H(z). -pi/. Fs/ Fs/ Fs/ Fs Frequency -pi Fs/ Fs/ Fs/ Fs Frequency

Chapter 5 Frequency Domain Analysis of Systems

Chapter 5 Frequency Domain Analysis of Systems CT, LTI Systems Consider the following CT LTI system: xt () ht () yt () Assumption: the impulse response h(t) is absolutely integrable, i.e., ht ( ) dt< (this