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Discrete-Time Systems Overview Useful Equations Let a k and b k be a sequence of complex numbers for k = 0 , 1 , . . . . Discrete-time systems Convolution y ( n ) h ( n ) x ( n ) y ( n ) h ( n ) x ( n )

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SLIDE 1

Discrete-Time Systems

H[x(n)] x(n) y(n)

  • System: any physical device or algorithm that transforms a signal

into another signal

  • The input signal, x(n), is sometimes called the excitation
  • The output signal produced by the system, y(n), is sometimes

called the response

  • The mathematical relationship between the input and output

signals is called the system model

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Discrete-Time Systems Overview

  • Discrete-time systems
  • Convolution
  • Properties
  • Systems with rational transfer functions
  • Correlation analysis
  • Minimum-phase and system invertibility
  • All-pass systems
  • Spectral factorization

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Linearity & Time Invariance Defined h(n)

x(n) y(n)

H(z)

x(n) y(n)

Consider any two bounded input signals x1(t) and x2(t). x1(n) → y1(n) x2(n) → y2(n)

  • Linear: A system is linear if and only if

a1x1(n) + a2x2(n) → a1y1(n) + a2y2(n) for any constant complex coefficients a1 and a2.

  • Time Invariant: A system is time invariant if and only if

x(n) → y(n) implies x(n − n0) → y(n − n0) for any signal x(n) and any integer n0.

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Useful Equations Let ak and bk be a sequence of complex numbers for k = 0, 1, . . . . y(n)

  • h(n) ∗ x(n)

y∗(n) = h∗(n) ∗ x∗(n) y(−n) = h(−n) ∗ x(−n) a−∗

k

  • 1

a∗

k

= 1 ak ∗

  • k

a∗

kb∗ k

=

  • k

akbk ∗

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SLIDE 2

LTI Systems: Convolution h(n)

x(n) y(n)

H(z)

x(n) y(n)

y(n) = x(n) ∗ h(n)

∞

  • k=−∞

x(k)h(n − k) =

∞

  • k=−∞

x(n − k)h(k)

  • Here ∗ denotes convolution
  • If x(n) and h(n) are finite duration, this can also be expressed in

matrix notation – Let x(n) be nonzero only for 0 ≤ n ≤ N − 1 – Let h(n) be nonzero only for 0 ≤ n ≤ M − 1 such that M < N – Then y(n) is nonzero only for 0 ≤ n ≤ L − 1 – L N + M − 1

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Linear Time-Invariant (LTI) Systems h(n)

x(n) y(n)

H(z)

x(n) y(n)

  • We shall limit our attention to linear systems
  • In most cases, we will assume the systems are also time-invariant

– This assumption can be relaxed with adaptive filters

  • Finally, we will assume the systems are always initially at rest
  • In this case, the impulse response completely characterizes the

system δ(n) → h(n)

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LTI Systems: Matrix Convolution

  • y(0)

y(1) . . . y(M − 2) y(M − 1) y(M) . . . y(N − 1) y(N) . . . y(L − 2) y(L − 1)

  • =
  • x(0)

· · · x(1) x(0) · · · . . . . . . ... . . . x(M − 2) x(M − 3) · · · x(M − 1) x(M − 2) · · · x(0) x(M) x(M − 1) · · · x(1) . . . . . . ... . . . x(N − 1) x(N − 2) · · · x(N − M) x(N − 1) · · · x(N − M + 1) . . . . . . ... . . . · · · x(N − 2) · · · x(N − 1)

  • h(0)

h(1) . . . h(M − 1)

  • y = Xh

X ∈ CL×M Note that all the elements along any diagonal of X are equal

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Justification of Scope

  • The assumption that a system is LTI and at rest may seem far too

restrictive

  • In practice these assumptions are rarely justified
  • Nonetheless, these concepts provide an important and practically

useful framework – Gives us a sense of what questions to ask about a system – LTI theory is the basis of many non-LTI techniques – In advanced courses nonlinear systems are sometimes treated as time-varying systems (varying operating points) – Time-varying systems are a “simple” generalization of LTI theory – Assumption can often be weakened to slowly time-varying – Basis of adaptive filters and moving-window approaches

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SLIDE 3

LTI Systems: Stability h(n)

x(n) y(n)

H(z)

x(n) y(n)

  • There are many definitions of stability
  • The most common is BIBO stability
  • BIBO Stability: A system is called bounded-input

bounded-output (BIBO) stable if every bounded input produces a bounded output |x(n)| < ∞ for all n → |y(n)| < ∞ for all n

  • An equivalent condition for LTI systems is

∞

  • n=−∞

|h(n)| < ∞

  • Note that this is a sufficient and necessary (proof?) condition for

the ROC of H(z) to include the unit circle

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LTI Systems: Matrix Convolution

  • y(0)

y(1) . . . y(M) y(M + 1) y(M + 2) . . . y(N − 2) y(N − 1) y(N) . . . y(L − 2) y(L − 1)

  • =
  • h(0)

· · · h(1) h(0) · · · . . . . . . ... . . . . . . h(M − 1) h(M − 2) · · · h(M − 1) · · · · · · . . . . . . ... . . . . . . · · · h(0) · · · h(1) h(0) · · · h(2) h(1) . . . . . . ... . . . . . . · · · h(M) h(M − 1) · · · h(M)

  • x(0)

x(1) . . . x(M) x(M + 1) x(M + 2) . . . x(N − 2) x(N − 1)

  • y = Hx

H ∈ CL×N Note that all the elements along any diagonal of X are equal

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LTI Systems: Causality h(n)

x(n) y(n)

H(z)

x(n) y(n)

  • Causal: A system in which y(n) depends only on present and/or

past values of x(n)

  • Note that we can implement non-causal discrete-time systems
  • ff-line if the entire input signal is known before the output must

be calculated

  • An LTI system is causal iff h(n) is causal:

h(n) = 0 for n < 0

  • This in turn implies that the ROC of H(z) includes |z| = ∞
  • Thus, LTI systems that are causal and stable have an ROC that

ranges from inside the unit circle to |z| = ∞

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LTI Systems: Matrix Convolution y = Xh y = Hx

  • X ∈ CL×M is called the input data matrix
  • It is a Toeplitz matrix since the diagonal elements are equal
  • Note also that the first and last M − 1 rows contain zero values

that are outside the range of x(n)

  • Thus the first and last M − 1 samples of y(n) contain boundary

effects

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SLIDE 4

LTI Systems: Rational Transfer Functions The transfer function can also be expressed as a product of terms due to each pole and zero H(z) = B(z) A(z) = G Q

k=1(1 − zkz−1)

P

k=1(1 − pkz−1)

  • Note that z is overloaded

– z−1 is the independent variable – zk is the kth zero of the system

  • Causal systems are stable, if all the poles are inside the unit

circle: |pk| < 1 for all k

  • Thus, the ROC is: |z| > maxk |pk|

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LTI Systems: Transform Domain Definitions h(n)

x(n) y(n)

H(z)

x(n) y(n)

We can also relate the DTFT and z-Transforms of x(n), y(n), and h(n) Y (z) = H(z)X(z) Y (ejω) = H(ejω)X(ejω)

  • Transfer function: H(z)

– Also called the system function

  • If the ROC of H(z) includes the unit circle, the system is stable

and H(ejω) exists

  • Frequency Response: H(ejω)
  • Magnitude Response: |H(ejω)|
  • Phase Response: ∠H(ejω)
  • Group Delay: − d

dω∠H(ejω) (not clear how to interpret) Portland State University ECE 538/638 Discrete-Time Systems

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LTI Systems: All-Zero Systems y(n) = −

P

  • k=1

aky(n − k) +

Q

  • k=0

bkx(n − k)

  • If P = 0, the equations simplify to

y(n) =

Q

  • k=0

bkx(n − k) h(n) =

  • bn

0 ≤ n ≤ Q

  • therwise
  • Finite Impulse Response (FIR): adjective for systems with a

finite impulse response: h(n) = 0 for all −∞ < N1 < n < N2 < ∞

  • Infinite Impulse Response (IIR): adjective for systems with an

impulse response that persists indefinitely

  • All-Zero Systems: systems with a finite impulse response

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LTI Systems: Difference Equations Many discrete-time systems can be described by constant-coefficient, linear difference equations: y(n) = −

P

  • k=1

aky(n − k) +

Q

  • k=0

bkx(n − k) The z-Transform of H(z) can then be expressed as H(z) = Y (z) X(z) = Q

k=0 bkz−k

1 + P

k=1 akz−k B(z)

A(z)

  • Note that I use bk for the coefficients of x(n − k) and the book

uses dk

  • If {ak, bk} depend on n (i.e. not constant), the system is

time-varying

  • If {ak, bk} depend on x(n) or y(n), then the system is nonlinear

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SLIDE 5

LTI Systems: All-Pole and Pole-Zero Systems H(z) = B(z) A(z) = G Q

k=1(1 − zkz−1)

P

k=1(1 − pkz−1)

  • All-Pole (AP): Systems with rational transfer functions and zeros
  • nly at z = 0
  • Pole-Zero (PZ): Systems with rational transfer functions, at

least one non-trivial zero (zk = 0), and at least one non-trivial pole (pk = 0)

  • Properties

– All-pole systems are IIR – Pole-zero systems are IIR – Not all IIR systems are pole-zero or all-pole – Example: h(n) = n−2u(n)

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FIR Filter Realization (direct form)

z-1 x(n) z-1 z-1 z-1

+ + + +

y(n) a0 a1 a2 a3 a4

  • There are many ways to implement filters
  • The above shows the direct-form realization
  • The realizations are all mathematically equivalent
  • Differ in

– Numerical stability (quantization) – Robustness to errors in filter coefficients

  • Will not discuss these issues in this class

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Example 1: Transform Domain Plot the impulse response, pole-zero diagram, frequency response, magnitude response, phase response, and group delay of a pole-zero causal system with the following poles at p1 = 0.5, p2,3 = −0.5 ± j0.8 and a zeros at z1,2 = 0.5 ± j0.8. The system has a DC gain of 10.

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LTI Systems: Exponential Components H(z) = B(z) A(z) = G Q

k=1(1 − zkz−1)

P

k=1(1 − pkz−1)

If we assume a rational transfer function has distinct poles (reasonable) and Q < P, we can express H(z) and h(n) as follows using partial fraction expansion H(z) =

P

  • k=1

Ak 1 − pkz−k h(n) =

P

  • k=1

Ak(pk)nu(n)

  • Each pole contributes an exponential component to h(n)
  • h(n) is causal (assumed)
  • h(n) is IIR
  • Rate of decay depends on how close |pk| is to one (the unit circle)

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SLIDE 6

Example 1: Pole-Zero Map

−1 −0.5 0.5 1 −1 −0.5 0.5 1

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Example 1: Impulse Response

10 20 30 40 50 60 70 −25 −20 −15 −10 −5 5 10 15 20 25 h(n) Time

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Example 1: MATLAB Code

h = circle; z = roots(b); p = roots(a); hold on; h2 = plot(real(z),imag(z),’bo’,real(p),imag(p),’rx’); hold off; axis square; xlim([-1.1 1.1]); ylim([-1.1 1.1]); axislines; box off; Portland State University ECE 538/638 Discrete-Time Systems

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Example 1: MATLAB Code

w = 0:0.01:pi; b = poly([0.5+j*0.8;0.5-j*0.8]); % Numerator a = poly([0.5;-0.5+j*0.8;-0.5-j*0.8]); % Denominator b = 10*b*sum(a)/sum(b); % Normalize to DC gain of 10 [H,w] = freqz(b,a,w); % Frequency response [G,w] = grpdelay(b,a,w); % Group delay figure n = 0:75; x = [1;zeros(max(n),1)]; y = filter(b,a,x); plot([min(n) max(n)],[0 0],’k:’); hold on; h = stem(n,y,’b’); set(h(1),’MarkerFaceColor’,’b’); set(h(1),’MarkerSize’,4); hold off; ylabel(’h(n)’); xlabel(’Time’); xlim([min(n) max(n)]); box off; Portland State University ECE 538/638 Discrete-Time Systems

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SLIDE 7

Example 1: Magnitude Response

0.5 1 1.5 2 2.5 3 100 200 |H(ejω)| 10

−2

10

−1

10 10 10

1

10

2

Frequency (rad/sample) |H(ejω)|

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Example 1: Frequency Response

0.5 1 1.5 2 2.5 3 −100 100 200 Real H(ejω) 0.5 1 1.5 2 2.5 3 −100 100 200 300 Frequency (rad/sample) Imaginary H(ejω)

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Example 1: MATLAB Code

subplot(2,1,1); h = plot(w,abs(H),’b’,’LineWidth’,1.5); ylabel(’|H(e^{j\omega})|’); box off; xlim([0 pi]); ylim([0 1.1*max(abs(H))]); subplot(2,1,2); h = loglog(w,abs(H),’b’,’LineWidth’,1.5); ylabel(’|H(e^{j\omega})| dB’); xlabel(’Frequency (rad/sample)’); box off; ylim([0.9*min(abs(H)) 1.1*max(abs(H))]); xlim([0.01 pi]); Portland State University ECE 538/638 Discrete-Time Systems

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Example 1: MATLAB Code

subplot(2,1,1); h = plot(w,real(H),’b’,’LineWidth’,1.5); ylabel(’Real H(e^{j\omega})’); box off; xlim([0 pi]); subplot(2,1,2); h = plot(w,imag(H),’b’,’LineWidth’,1.5); ylabel(’Imaginary H(e^{j\omega})’); xlabel(’Frequency (rad/sample)’); box off; xlim([0 pi]); Portland State University ECE 538/638 Discrete-Time Systems

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SLIDE 8

LTI Frequency Domain Analysis H(z)

x(n) y(n) Finite Energy Periodic

π π π π π π ω ω ω ω ω ω |X(ejω)| |H(ejω)| |H(ejω)| |Y (ejω)| |ak| |akH(ejkω0)|

  • Although we treat energy and periodic signals differently, LTI

systems affect the signals in a similar manner

  • If we relax the DTFT to allow unit-impulses, δ(ω), the DTFT can

be used to analyze some power signals (energy signal + periodic signals) as well as energy signals

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Example 1: Phase & Group Delay

0.5 1 1.5 2 2.5 3 −1 1 2 3 Phase (radians) 0.5 1 1.5 2 2.5 3 −20 −10 10 20 Frequency (rad/sample) Group Delay (samples)

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Correlation Sequence Correlation sequence between two discrete-time signals x(n) and y(n). If x(n) and y(n) are energy signals (E∞ < ∞), then rxy(ℓ)

∞

  • n=−∞

x(n)y∗(n − ℓ) If x(n) and y(n) are power signals (0 < P∞ < ∞), then rxy(ℓ) lim

N→∞

1 2N + 1

N

  • n=−N

x(n)y∗(n − ℓ)

  • Is a measure of quantitative similarity between two signals
  • ℓ is called the lag or shift
  • Is not statistical correlation (signals are deterministic)

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Example 1: MATLAB Code

subplot(2,1,1); h = plot(w,angle(H),’b’,’LineWidth’,1.5); ylabel(’Phase (radians)’); box off; xlim([0 pi]); subplot(2,1,2); h = plot(w,G,’b’,’LineWidth’,1.5); ylabel(’Group Delay (samples)’); xlabel(’Frequency (rad/sample)’); box off; xlim([0.01 pi]); Portland State University ECE 538/638 Discrete-Time Systems

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SLIDE 9

Autocorrelation Sequence for Power Signals If x(n) is a power signal (0 < P∞ < ∞), then rx(ℓ) lim

N→∞

1 2N + 1

N

  • n=−N

x(n)x∗(n − ℓ)

  • Recall that the DTFT was not defined for power signals
  • The DTFS was defined for power signals, but only periodic power

signals

  • The autocorrelation sequence is defined for both energy and power

signals

  • The power signals need not be periodic!

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Autocorrelation Sequence for Energy Signals Autocorrelation sequence: the correlation sequence between x(n) and itself. If x(n) is an energy signal (E∞ < ∞), then rxx(ℓ)

∞

  • n=−∞

x(n)x∗(n − ℓ)

  • This is often denoted with simplified subscripts,

rxx(ℓ) = rx(ℓ) = r(ℓ), when clear from context.

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Energy/Power Spectral Density Energy/Power Spectral Density: the DTFT of the autocorrelation sequence rx(ℓ) Rx(ejω) = F {rx(ℓ)} =

∞

  • n=−∞

rx(n)e−jωn

  • You should be able to show that for energy signals

Rx(ejω) = |X(ejω)|2

  • The name comes from Parseval’s relation, as discussed earlier
  • If rx(n) has complex conjugate symmetry, Rx(ejω) is real and the

complex phase angle is zero at all frequencies

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Autocorrelation Symmetry rx(ℓ) has complex-conjugate symmetry rx(ℓ) =

∞

  • n=−∞

x(n)x∗(n − ℓ) =

  • ∞
  • n=−∞

x∗(n)x(n − ℓ) ∗ m

  • n − ℓ

rx(ℓ) =

  • ∞
  • m=−∞

x∗(m + ℓ)x(m) ∗ =

  • ∞
  • m=−∞

x(m)x∗(m + ℓ) ∗ rx(ℓ) = r∗

x(−ℓ)

If x(n) is real, rx(ℓ) is an even signal: rx(ℓ) = rx(−ℓ)

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SLIDE 10

Correlation Analysis of LTI Systems: Time Domain You should be able to show the following are true rx(ℓ) = r∗

x(−ℓ)

ryx(ℓ) = h(ℓ) ∗ rx(ℓ) = r∗

xy(−ℓ)

rxy(ℓ) = h∗(−ℓ) ∗ r∗

x(−ℓ)

= h∗(−ℓ) ∗ rx(ℓ) ry(ℓ) = h(ℓ) ∗ rxy(ℓ) = h(ℓ) ∗ r∗

yx(−ℓ)

= h∗(−ℓ) ∗ ryx(ℓ) = h∗(−ℓ) ∗ r∗

xy(−ℓ)

= rh(ℓ) ∗ rx(ℓ) rh(ℓ)

  • ∞
  • n=−∞

h(n)h∗(n − ℓ) = h(ℓ) ∗ h∗(−ℓ)

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Example 2: Convergence Show that Rx(ejω) = |X(ejω)|2 where x(n) is an energy signal. Hint: recall x(−n)

Z

← → X(z−1) x∗(n)

Z

← → X∗(z∗)

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Correlation Analysis of LTI Systems: Frequency Domain The equations on the previous slide become the following in the z domain: Ryx(z) = H(z)Rx(z) Ry(z) = H∗(z−∗)Ryx(z) = Rh(z)Rx(z) Rh(z)

  • H(z)H∗(z−∗)

where z−∗

1 z∗ =

1

z

∗. If h(n) is real, the last expression simplifies to Rh(z) = H(z)H(z−1) On the unit circle, z = ejω and H(e−jω) = H∗(ejω), so this becomes Ry(ejω) = |H(ejω)|2Rx(ejω) Rh(ejω) = |H(ejω)|2

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Example 2: Workspace

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SLIDE 11

System Invertibility: Frequency Domain h(n)

x(n) y(n)

hi(n)

x(n)

In the frequency domain, this becomes trivial Hi(z) = 1 H(z) If H(z) is a pole-zero system, we have H(z) = B(z) A(z) Hi(z) = A(z) B(z)

  • Even if the ROC is specified for H(z), the inverse is not unique
  • The ROC for H(z) does not specify the ROC for Hi(z)
  • We must know or assume additional information about Hi(z) for

the inverse to be unique

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All-Pass Signals Rx(ejω) = |X(ejω)|2 = G2 − π < ω ≤ π

  • All-pass signals are signals with a flat (constant) energy spectral

density

  • Note that this is the definition of a signal type. Should not be

confused with all-pass systems (to be defined later)

  • Note too that this definition applies to deterministic signals
  • Example: x(n) = δ(n − n0) for any constant n0
  • The output of an LTI system to an all-pass signal is then

ry(ℓ) = rh(ℓ) ∗ rx(ℓ) = G2rh(ℓ) Ry(z) = H∗(z−∗)H(z)Rx(z) = G2Rh(z)

  • Only constrains magnitude of X(ejω), not the phase

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Minimum-Phase Systems h(n) = 0 n < 0

∞

  • n=0

|h(n)| < ∞ hi(n) = 0 n < 0

∞

  • n=0

|hi(n)| < ∞

  • Minimum-Phase System: A system for which the system

impulse response h(n) and an inverse system hi(n) are both causal and stable

  • If a system is minimum-phase, its inverse is minimum-phase (by

definition)

  • Causal PZ systems are minimum-phase if all poles and zeros are

inside the unit circle: |pk| < 1 and |zk| < 1 for all k

  • Minimum-phase systems have the smallest group delay of all

causal, stable systems with the same magnitude response (minimum group delay systems might be a better name)

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System Invertibility: Time Domain h(n)

x(n) y(n)

hi(n)

x(n)

  • Invertible System a system in which the input signal can be

uniquely determined from the system’s output signal

  • Implies there exists a second system that produces the output

signal from the input: y(n) → x(n)

  • The cascade of the system, H and the inverse system, Hi, is the

identity system

  • If a system is LTI and its inverse exists, the inverse is also LTI.

Thus, [x(n) ∗ h(n)] ∗ hi(n) = x(n) h(n) ∗ hi(n) = δ(n)

  • Solving this last equation for hi(n) directly is generally difficult to

solve and may have multiple solutions

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SLIDE 12

All-Pass Systems: Poles and Zeros

Im Re 1

The poles and zeros of PZ all-pass systems are complex conjugate reciprocals of one another pk = p0 → zk = 1 p∗ zk = z0 → pk = 1 z∗

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Minimum-Phase Terms

  • Maximum-Phase System: A causal, stable system with

maximum group delay of all causal, stable systems with the same magnitude response

  • The book has this wrong. Please see the errata on the course web

site.

  • Mixed-Phase System: A system that is neither minimum-phase
  • r maximum-phase
  • This is all important because, if h(n) is real,

Ry(z) = H∗(z−∗)H(z)Rx(z) = H(z)H(z−1)Rx(z)

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Selected All-Pass System Properties

  • The output energy of a stable all-pass system is equal to the input

energy Ey =

∞

  • k=−∞

|y(n)|2 = 1 2π π

−π

|Y (ejω)|2 dω = Ex

  • A causal, stable, PZ, all-pass system has a phase response that

decreases monotonically

  • All-pass systems have nonnegative group delay

Other properties are listed in the text.

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All-Pass Systems All-pass System: An LTI system with a unit magnitude response |Hap(ejω)| = 1 − π < ω ≤ π

  • Example: H(z) = zk

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SLIDE 13

Spectral Factorization: Why Minimum-Phase h(n)

x(n) y(n)

H(z)

x(n) y(n)

  • Why assume the system is minimum-phase?
  • Because minimum-phase systems have desirable properties

– System is causal and stable (i.e. can be implemented in real time) – Inverse is causal and stable – System is invertible

  • In most instances, we have no cause to pick any other solution
  • Why pick a mixed-phase or maximum phase system?

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Pole-Zero Systems Revisited Hmin(z)

x(n) y(n)

Hap(z) H(z) Any causal, stable PZ system can be expressed as H(z) = Hmin(z)Hap(z)

  • Since causal and stable, we know all of the poles are inside the

unit circle

  • Any zero that is outside the unit circle can be “moved inside” by

the all-pass system

  • Note that the magnitude response of H(z) is the same as that of

Hmin(z)

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Spectral Factorization: Solving for H(z) Ry(ejω) = |H(ejω)|2Rx(ejω) = G2|H(ejω)|2

  • One approach to spectral factorization is to solve for the roots of

Ry(z)

  • For every pole/zero inside the unit circle, there is another at the

conjugate reciprocal

  • Thus we simply pick all of the poles and zeros inside the unit circle
  • Every rational PSD has, to within a scale factor, a unique

minimum-phase factorization

  • There are 2P +Q factorizations with the same PSD, but only one is

minimum-phase

  • Not all rational functions of z are valid spectral densities: Ry(z)

– The roots of the numerator and denominator must occur in conjugate reciprocals: zk and

1 z∗

k

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Spectral Factorization: Introduction h(n)

x(n) y(n)

H(z)

x(n) y(n)

  • Later this term we will discuss estimators of ry(ℓ)
  • If we assume the input signal is all-pass, rx(ℓ) = G2δ(ℓ), can we

determine the transfer function H(ejω)?

  • Recall

Ry(ejω) = |H(ejω)|2Rx(ejω) = G2|H(ejω)|2

  • Thus, even if Rx(ejω) and Ry(ejω) are known, estimated,

assumed, or specified, H(ejω) is not uniquely specified

  • Only the magnitude of H(ejω) is specified, not the phase
  • If we assume H(ejω) is minimum-phase, the solution is unique
  • The process of solving for Hmin(z) from Ry(z) with Rx(z) = G2

is called spectral factorization

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SLIDE 14

Spectral Factorization: General Theorem If (1) the input signal to a system, x(n), is all-pass signal such that Rx(z) = G2, (2) ln Ry(z) is analytic in an open ring α < |z| < 1

α in

the z-plane, and (3) the ring includes the unit circle, then Ry(z) can be factored as Ry(z) = G2Hmin(z)H∗

min(z−∗)

where Hmin(z) is a minimum-phase system.

  • This is more general than the spectral factorization in most texts:

we do not require that Ry(z) be a rational function of z

  • Paley-Wiener Theorem: The spectral factorization is possible

(ln Ry(z) is analytic on the unit circle) if Ry(z) satisfies the Paley-Winer condition π

−π

  • ln Ry(ejω)
  • dω < ∞

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Spectral Factorization: Summary Ry(z) = G2Hmin(z)H∗

min(z−∗)

  • In general, spectral factorization is hard
  • If Ry(z) is rational, it is easy
  • We will discuss estimators of Ry(z) for stochastic signals later this

term

  • If the input signal is Gaussian, Ry(z) and the signal mean are a

complete statistical description of the signal

  • In this case, we cannot tell from the signal properties whether the

signal was produced by a minimum-phase system or a mixed phase system

  • If there is domain knowledge that enables us to favor one model,

we should pick the system with the most favorable properties: Hmin(z)

  • More on this later

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