Amplitude Modulation X1(w) Y1(w) y1(t) = x1(t) cos(wc t) cos(wc t) 1 w1 -w1 X1(w) Y1(w) ½ -wc - w1 -wc + w1 -wc wc - w1 wc + w1 wc ½X1(w - wc) ½X1(w + wc) lower sidebands cos(wc t) x1(t) The bandwidth just doubled!! How can we compensate for the bandwidth cost? y1(t)

Bandwidth Efficiency X2(w) Y2(w) + cos(c t) sin(c t) x1(t) x2(t) y2(t) = x2(t) sin(wc t) w 1 w2 -w2 X2(w) Y2(w) j ½ -wc – w2 -wc + w2 -wc wc – w2 wc + w2 wc -j ½X2(w - wc) j ½X2(w + wc) -j ½ lower sidebands BW X1(t) = w1 BW Y1(t) = 2*w1 + cos(c t) sin(c t) x1(t) x2(t) s(t) Real BW S(t) = 2*max (w1,w2) Real and imaginary dimensions are orthogonal Imaginary BW X2(t) = w2 BW Y2(t) = 2*w2

Signals & Systems Systems Signals Linear Time Invariant Additivity: x1[n] + x2[n]  y1[n] + y2[n] Homogeneity: a x[n]  a y[n] for any real/complex constant a Linear Systems Time Invariant x[n - m]  y[n - m], for all m Signals

FIR Delay = - slope = N/2 N = order of the FIR filter NOT all FIR have linear phase Phase FIR filter must be with impulse response that is symmetric or anti-symmetric about its midpoint LTI system Frequency Response Delay = - slope = N/2 N = order of the FIR filter First Order Filter

IIR Feedback Feedforward

Design Considerations IIR Difference Equation Transfer Function N zeros & M poles Stability A causal infinite impulse response LTI system is BIBO stable if and only if its poles lie inside the unit circle If poles were on the unit circle then the filter impulse response will be an oscillator Design Considerations Filter of order n will have n/2 conjugate poles if n is even OR one real root and (n-1)/2 conjugate poles if n is odd Cascade biquads from input to output in order of ascending quality factors (first-order section has lowest quality factor) For each biquad, choose conjugate zeros closest to pole pair

Conclusion FIR Filters IIR Filters Implementation complexity (1) Higher Lower (sometimes by factor of four) Minimum order design Parks-McClellan (Remez exchange) algorithm (2) Elliptic design algorithm Stable? Always May become unstable when implemented (3) Linear phase If impulse response is symmetric or anti-symmetric about midpoint No, but phase may made approximately linear over passband (or other band) (1) For same piecewise constant magnitude specification (2) Algorithm to estimate minimum order for Parks-McClellan algorithm by Kaiser may be off by 10%. Search for minimum order is often needed. (3) Algorithms can tune design to implementation target to minimize risk

IIR Magnitude Response Re(z) Im(z) X O |ej – p0| is distance from point on unit circle ej and pole p0

Pole Zero Plot What is the frequency selectivity of this filter? Im(z) Re(z) Im(z) O X

Exercise Filter type? Difference equation? Block Diagram? Initial conditions? Frequency response? Frequency selectivity? Value of C to normalize the magnitude response?

IIR – FIR How many poles and zeros? 5 positive zeros, but the phase is linear so we have other 5 negative zeros. Then order is >= 10 Slope = group delay = order/2

Modulation Demodulation cos(wct) m(t) s(t) How did this term disappear ?!!