Discrete-Time LTI Systems and Analysis Excitation Response Discrete-Time Discrete-Time Signal Signal Dr

Discrete-Time LTI Systems Discrete-time Systems Input-Output Description of Dst-Time Systems

x(n) y(n) input/ Discrete-time output/ System Discrete-Time LTI Systems and Analysis excitation response Discrete-time Discrete-time signal signal Dr. Deepa Kundur

University of Toronto I Input-output description (exact structure of system is unknown or ignored): y(n) = T [x(n)]

I “black box” representation:

x(n) −→T y(n)

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Discrete-Time LTI Systems Discrete-time Systems Discrete-Time LTI Systems Discrete-time Systems System Properties Terminology: Implication

If“ A” then“ B” Shorthand: A =⇒ B Why is this so important? Example 1: I mathematical techniques developed to analyze systems are often it is snowing= ⇒ it is at or below freezing temperature contingent upon the general characteristics of the systems being Example 2: considered α ≥ 5.2= ⇒ α is positive Note: For both examples above, B 6=⇒ A I for a system to possess a given property, the property must hold for every possible input to the system

I to disprove a property, need a single counter-example

I to prove a property, need to prove for the general case

Dr. Deepa Kundur (University of Toronto) Discrete-Time LTI Systems and Analysis 3 / 61 Dr. Deepa Kundur (University of Toronto) Discrete-Time LTI Systems and Analysis 4 / 61 Discrete-Time LTI Systems Discrete-time Systems Discrete-Time LTI Systems Discrete-time Systems Terminology: Equivalence Common Properties Time-invariant system: input-output characteristics do not If“ A” then“ B” Shorthand: A =⇒ B I change with time

and I a system is time-invariant iﬀ If“ B” then“ A” Shorthand: B =⇒ A T T x(n) −→ y(n) =⇒ x(n − n0) −→ y(n − n0) can be rewritten as for every input x(n) and every time shift n0. “A” if and only if“ B” Shorthand: A ⇐⇒ B

We can also say:

I A is EQUIVALENT to B =

I A = B

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Discrete-Time LTI Systems Discrete-time Systems Discrete-Time LTI Systems Discrete-time Systems Common Properties Additivity:

I Linear system: obeys superposition principle

I a system is linear iﬀ

T [a1 x1(n) + a2 x2(n)] = a1 T [x1(n)] + a2 T [x2(n)]

for any arbitrary input sequences x1(n) and x2(n), and any arbitrary constants a1 and a2.

Homogeneity:

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I Causal system: output of system at any time n depends only on present and past inputs

I a system is causal iﬀ Recall: y(n) = F [x(n), x(n − 1), x(n − 2),...] ∞ X for all n. x(n) = x(k)δ(n − k) k=−∞ I Bounded Input-Bounded output (BIBO) Stable: every bounded input produces a bounded output

I a system is BIBO stable iﬀ

|x(n)| ≤ Mx < ∞ =⇒ |y(n)| ≤ My < ∞

for all n and for all possible bounded inputs.

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Discrete-Time LTI Systems The Convolution Sum Discrete-Time LTI Systems The Convolution Sum The Convolution Sum The Convolution Sum Let the response of a linear time-invariant (LTI) system denoted T to the unit sample input δ(n) be h(n).

T Therefore, δ(n) −→ h(n) T ∞ δ(n − k) −→ h(n − k) X y(n) = x(k)h(n − k) = x(n) ∗ h(n) T αδ (n − k) −→ α h(n − k) k=−∞ x(k) δ(n − k) −→T x(k) h(n − k) ∞ ∞ for any LTI system. X T X x(k)δ(n − k) −→ x(k)h(n − k) k=−∞ k=−∞ x(n) −→T y(n)

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For a causal system, y(n) only depends on present and past inputs values. Therefore, for a causal system, we have: It can also be shown that ∞ X ∞ y(n) = h(k)x(n − k) X |h(n)| < ∞ ⇐⇒ LTI system is BIBO stable k=−∞ n=−∞ −1 ∞ X X = h(k)x(n − k)+ h(k)x(n − k) Note: k=−∞ k=0 ∞ I ⇐⇒ means that the two statements are equivalent X = h(k)x(n − k) I BIBO = bounded-input bounded-output k=0 where h(n) = 0 for n < 0 to ensure causality.

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Discrete-Time LTI Systems The Convolution Sum Discrete-Time LTI Systems The Convolution Sum

PROOF PROOF

For a stable system, y(n) is bounded if x(n) is bounded. What are the implications on h(n)? P∞ We have: To prove the reverse implication (i.e., necessity), assuming n=−∞ |h(n)| = ∞ we must ﬁnd a ∞ bounded input x(n) that will always result in an unbounded y(n). Recall, X |y(n)| = | h(k)x(n − k)| ∞ k=−∞ X ∞ ∞ y(n) = h(k)x(n − k) X X ≤ |h(k)x(n − k)| = |h(k)|·| x(n − k)| k=−∞ ∞ ∞ k=−∞ k=−∞ | {z } X X |x(n)|≤Mx <∞ y(0) = h(k)x(0 − k) = h(k)x(−k) ∞ ∞ X X k=−∞ k=−∞ ≤ |h(k)|Mx = Mx |h(k)| k=−∞ k=−∞ Consider x(n) = sgn(h(−n)); note: |x(n)| ≤ 1. ∞ Therefore, P∞ |h(k)| < ∞ is a suﬃcient condition to guarantee: X k=−∞ y(0) = h(k)x(−k) k=−∞ ∞ X ∞ ∞ y(n) ≤ Mx |h(k)| < ∞ X X = h(k)sgn(h(−(−k)))= h(k)sgn(h(k)) k=−∞ k=−∞ k=−∞ ∞ and we can write: X ∞ = |h(n)| = ∞ X |h(n)| < ∞ =⇒ LTI system is stable n=−∞ n=−∞

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PROOF The Direct z-Transform

Therefore, ∞ X |h(n)| = ∞ I Direct z-Transform: n=−∞ ∞ guarantees that there exists a bounded input that will result in an unbounded output, so it is X −n also a necessary condition and we can write: X (z) = x(n)z n=−∞ ∞ X |h(n)| < ∞⇐ = LTI system is stable n=−∞ I Notation:

Putting suﬃciency and necessity together we obtain: X (z) ≡ Z{x(n)}

∞ X |h(n)| < ∞ ⇐⇒ LTI system is stable Z n=−∞ x(n) ←→ X (z)

Note: ⇐⇒ means that the two statements are equivalent.

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Discrete-Time LTI Systems The z-Transform and System Function Discrete-Time LTI Systems The z-Transform and System Function Region of Convergence z-Transform Properties

Property Time Domain z-Domain ROC Notation: x(n) X (z) ROC: r2 < |z| < r1 x1(n) X1(z) ROC1 I the region of convergence (ROC) of X (z) is the set of all values x2(n) X1(z) ROC2 Linearity: a1x1(n) + a2x2(n) a1X1(z) + a2X2(z) At least ROC1∩ ROC2 of z for which X (z) attains a ﬁnite value Time shifting: x(n − k) z−k X (z) ROC, except z = 0 (if k > 0) and z = ∞ (if k < 0) n −1 I The z-Transform is, therefore, uniquely characterized by: z-Scaling: a x(n) X (a z) |a|r2 < |z| < |a|r1 Time reversal x(−n) X (z−1) 1 < |z| < 1 1. expression for X (z) r1 r2 Conjugation: x∗(n) X ∗(z∗) ROC 2. ROC of X (z) dX (z) z-Diﬀerentiation: n x(n) −z dz r2 < |z| < r1 Convolution: x1(n) ∗ x2(n) X1(z)X2(z) At least ROC1∩ ROC2

among others . . .

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Signal, x(n) z-Transform, X (z) ROC 1 δ(n) 1 All z h(n) ←→Z H(z) 1 2 u(n) 1−z−1 |z| > 1 Z n 1 time-domain ←→ z-domain 3 a u(n) −1 |z| > |a| 1−az Z n az−1 impulse response ←→ system function 4 na u(n) (1−az−1)2 |z| > |a| n 1 5 −a u(−n − 1) 1−az−1 |z| < |a| n az−1 Z 6 −na u(−n − 1) (1−az−1)2 |z| < |a| y(n) = x(n) ∗ h(n) ←→ Y (z) = X (z) · H(z) −1 1−z cos ω0 7 (cos(ω0n))u(n) −1 −2 |z| > 1 1−2z cos ω0+z −1 z sin ω0 8 (sin(ω0n))u(n) −1 −2 |z| > 1 1−2z cos ω0+z −1 n 1−az cos ω0 Therefore, 9 (a cos(ω0n)u(n) 2 −2 |z| > |a| 1−2az−1 cos ω0+a z −1 Y (z) n 1−az sin ω0 H(z) = 10 (a sin(ω0n)u(n) 2 −2 |z| > |a| 1−2az−1 cos ω0+a z X (z)

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Discrete-Time Fourier Analysis DTFT Discrete-Time Fourier Analysis DTFT Discrete-Time Fourier Transform (DTFT) Periodicity of the DTFT

Consider

I DTFT pair: ∞ X Z X (ω + 2π) = x(n)e−j(ω+2π)n 1 jωn x(n) = X (ω)e dω n=−∞ 2π 2π ∞ ∞ X X = x(n)e−jωn · e−j2πn X (ω) = x(n)e−jωn n=−∞ n=−∞ ∞ ∞ X X = x(n)e−jωn · 1 = x(n)e−jωn = X (ω) I X (ω) is the decomposition of x(n) into its frequency components. n=−∞ n=−∞ Therefore, X (ω) is periodic with a period of 2π.

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I Continuous-Time Sinusoids: Frequency and Rate of Oscillation: I Since X (ω) = X (ω + 2π), when dealing with discrete frequencies, only a continuous frequency range of length 2π x(t) = A cos(ωt + φ) (representing one period) needs to be considered.

I Minimum frequency for ω = 2kπ, k ∈ Z 2π 1 I Maximum frequency for ω = (2k + 1)π, k ∈ Z T = = I Convention is to use ω ∈ [0, 2π) or ω ∈ (−π, π] ω f

I Frequency range of a discrete-time signal is considered to be ω ∈ (−π, π] Rate of oscillation increases as ω increases (or T decreases).

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Discrete-Time Fourier Analysis DTFT Discrete-Time Fourier Analysis DTFT ω smaller ω larger, rate of oscillation higher

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MINIMUM OSCILLATION

I Discrete-Time Sinusoids: Frequency and Rate of Oscillation: x[n] = A cos(Ωn + φ)

Rate of oscillation increases as Ω increases UP TO A POINT then MAXIMUM OSCILLATION decreases again and then increases again and then decreases again ....

MINIMUM OSCILLATION

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Discrete-Time Fourier Analysis DTFT Discrete-Time Fourier Analysis DTFT

-2 2 6 -2 2 6 -3 -1 0 1 3 4 5 -3 -1 0 1 3 4 5

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jΘ(ω) among others . . . H(ω) = |H(ω)|e |H(ω)| ≡ system gain for freq ω ∠H(ω) = Θ(ω) ≡ phase shift for freq ω

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Discrete-Time LTI Filtering Discrete-Time LTI Filtering Complex Nature of X (jω) Complex Nature of X (jω)

Recall, Fourier Transform: Z ∞ −jωt I Rectangular coordinates: rarely used in signal processing X (jω) = x(t)e dt ∈ C −∞ X (jω)= XR (jω)+ jX I (jω) and Inverse Fourier Transform: where XR (jω), XI (jω) ∈ R. 1 Z ∞ x(t) = X (jω)ejωt dω 2π −∞ 1 Z 0 1 Z ∞ = X (jω)ejωt dω + X (jω)ejωt dω I Polar coordinates: more intuitive way to represent frequency content 2π −∞ 2π 0 X (jω)= |X (jω)| ej∠X (jω) Note: If x(t) is real, then the imaginary part of the negative frequency sinusoids where |X (jω)|, ∠X (jω) ∈ R. (i.e., ejωt for ω<0) cancel out the imaginary part of the positive frequency sinusoids (i.e., ejωt for ω>0)

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1 Z ∞ x(t) = X (jω)ejωt dω 2π −∞ jωt ∞ I |X (jω)|: determines the relative presence of a sinusoid e in 1 Z = |X (jω)| ej∠X (jω)ejωt dω x(t) 2π −∞ 1 Z ∞ = |X (jω)| ej(ωt+∠X (jω))dω 2π −∞

I X (jω): determines how the sinusoids line up relative to one ∠ j(ωt+ X (jω)) another to form x(t) I Recall, e ∠ = cos(ωt + ∠X (jω)) + j sin(ωt + ∠X (jω)). jωt I The larger |X (jω)| is, the more prominent e is in forming x(t).

I ∠X (jω) determines the relative phases of the sinusoids (i.e. how they line up with respect to one another).

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Discrete-Time LTI Filtering Discrete-Time LTI Filtering LTI Filtering LTI Systems as Frequency-Selective Filters

∞ X y(n) = x(k)h(n − k) k=−∞ I Filter: device that discriminates, according to some attribute of Y (ω) = H(ω)X (ω) the input, what passes through it where F x(n) ←→ X (ω) I For LTI systems, given Y (ω) = H(ω)X (ω) h(n) ←→F H(ω) I H(ω) acts as a weighting or spectral shaping function of the F y(n) ←→ Y (ω) diﬀerent frequency components of the signal

I LTI system is known as a frequency shaping ﬁlter

H(ω)= |H(ω)|ejΘ(ω) LTI system ⇔ ﬁlter |Y (ω)| = |H(ω)||X (ω)| ∠Y (ω)=Θ( ω)+ ∠X (ω)

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Definition: a discrete-time finite impulse response (FIR) filter is one Definition: a discrete-time infinite impulse response (IIR) filter is one in which the associated impulse response has finite duration. in which the associated impulse response has infinite duration. ∞ X ∞ y(n) = h(k)x(n − k) X y(n) = h(k)x(n − k) k=−∞ k=−∞ M−1 ∞ X X = h(k)x(n − k) = h(k)x(n − k) k=0 k=0

I lower limit of k = 0 is from causality requirement I lower limit of k = 0 is from causality requirement I upper limit of 0 ≤ M − 1 < ∞ is from the ﬁnite duration I necessary upper limit of ∞ is from the inﬁnite duration requirement; in this case the support is M consecutive points requirement starting at time 0 and ending at M − 1

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Discrete-Time LTI Filtering Discrete-Time LTI Filtering LCCDEs LCCDEs Linear constant coeﬃcient diﬀerence equations (LCCDEs) are an Q: Why does an LCCDE have a rational system function?

important class of ﬁlters that we consider in this course: N M X X y(n) = − ak y(n − k) + bk x(n − k) k=1 k=0 N M N M X X X X y(n) = − ak y(n − k) + bk x(n − k) a0y(n) = − ak y(n − k) + bk x(n − k) a0 ≡ 1 k=1 k=0 k=1 k=0 N M X X ak y(n − k) = bk x(n − k) They have a rational system function: k=0 k=0 N M PM −k X X b z polynomial in z Z{ ak y(n − k)} = Z{ bk x(n − k)} H(z) = k=0 k = PN −k another polynomial in z k=0 k=0 1 + k=1 ak z N M X X ak Z{y(n − k)} = bk Z{x(n − k)} Depending on the values of N, M, ak and bk they can correspond to k=0 k=0 either FIR or IIR ﬁlters.

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N M Property Time Domain z-Domain ROC X X Notation: x(n) X (z) ROC: r2 < |z| < r1 ak Z{y(n − k)} = bk Z{x(n − k)} x1(n) X1(z) ROC1 k=0 | {z } k=0 | {z } z−k Y (z) z−k X (z) x2(n) X1(z) ROC2 Linearity: a1x1(n) + a2x2(n) a1X1(z) + a2X2(z) At least ROC1∩ ROC2 N M Time shifting: x(n − k) z−k X (z) ROC, except X −k X −k ak z Y (z) = bk z X (z) z = 0 (if k > 0) and z = ∞ (if k < 0) k=0 k=0 n −1 N M z-Scaling: a x(n) X (a z) |a|r2 < |z| < |a|r1 −1 1 1 X −k X −k Time reversal x(−n) X (z ) < |z| < Y (z) ak z = X (z) bk z r1 r2 Conjugation: x∗(n) X ∗(z∗) ROC k=0 k=0 dX (z) M z-Diﬀerentiation: n x(n) −z dz r2 < |z| < r1 P −k Y (z) k=0 bk z Convolution: x1(n) ∗ x2(n) X1(z)X2(z) At least ROC1∩ ROC2 H(z) ≡ = a0 ≡ 1 X (z) PN −k k=0 ak z among others . . . PM b z−k PM b z−k H(z) = k=0 k = k=0 k −0 PN −k PN −k 1 · z + k=1 ak z 1 + k=1 ak z

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Discrete-Time LTI Filtering Discrete-Time LTI Filtering FIR LCCDEs Block Diagram Represenation

M−1 ∞ X X y(n) = bk x(n − k) = h(k)x(n − k) Adder: + k=0 k=−∞ M−1 Unit delay: X −k H(z) = bk z k=0 Constant multiplier: Please note: upper limit is M − 1 opposed to M (which is used for the general Unit advance: LCCDE case) to meet common FIR convention of an M-length ﬁlter. By inspection: b 0 ≤ n ≤ M − 1 h(n) = n Signal multiplier: + 0 otherwise

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M−1 X y(n) = bk x(n − k) N M k=0 X X y(n) = − ak y(n − k) + bk x(n − k) ... k=1 k=0 PM −k M k=0 bk z X −k 1 H(z) = = bk z · PN −k PN −k 1 + k=1 ak z k=0 1 + k=1 ak z | {z } | {z } H (z) + + + ... + + H1(z) 2 = H1(z) · H2(z) Requires:

I M multiplications

I M − 1 additions

I M − 1 memory elements

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Discrete-Time LTI Filtering Discrete-Time LTI Filtering Adder: + Direct Form I IIR Filter Implementation Direct Form II IIR Filter Implementation Unit delay: + + + +

Constant multiplier: + + + +

+ +Unit advance: + +

+ + + + Signal multiplier: + ...... + + + +

LTI All-zero system LTI All-pole system LTI All-pole system LTI All-zero system

Requires: M + N + 1 multiplications, M + N additions, M + N memory locations Requires: M + N + 1 multiplications, M + N additions, M + N memory locations

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Adder: + Adder: +

Unit delay: Unit delay:

Constant multiplier: Constant multiplier:

Unit advance: Unit advance:

Signal multiplier: + Signal multiplier: + Discrete-Time LTI Filtering Discrete-Time LTI Filtering Direct Form II IIR Filter Implementation Stability of Rational System Function Filters

+ +

+ + N M X X + + y(n) = − ak y(n − k) + bk x(n − k) k=1 k=0 PM b z −k + + H(z) = k=0 k PN −k ...... 1 + k=1 ak z + + Recall, for BIBO stability of a causal system the system poles must be strictly inside the unit circle.

... For N>M Why? Requires: M + N + 1 multiplications, M + N additions, max(M, N) memory locations Dr. Deepa Kundur (University of Toronto) Discrete-Time LTI Systems and Analysis 53 / 61 Dr. Deepa Kundur (University of Toronto) Discrete-Time LTI Systems and Analysis 54 / 61

Discrete-Time LTI Filtering Discrete-Time LTI Filtering Adder: + Stability of Rational System Function Filters Stability of Rational System Function Filters Unit delay:

∞ Constant multiplier: X H(z) = h(n)z −n Unit advance: n=−∞ Recall, ∞ ∞ X −n X −n ∞ |H(z)| ≤ |h(n)z | = |h(n)||z | Signal multiplier:X + |h(n)| < ∞ ⇐⇒ LTI system is stable n=−∞ n=−∞ n=−∞ When evaluated for |z| = 1 (i.e., on the unit circle),

∞ X |H(z)| ≤ |h(n)|< ∞ n=−∞

Therefore, BIBO stability =⇒ ROC includes unit circle ROC includes unit circle =⇒ BIBO stability is also true.

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Therefore,

∞ H(z) ROC X LTI system |h(n)| < ∞ ⇐⇒ ⇐⇒ includes is stable n=−∞ unit circle For a causal rational system function, the ROC includes the unit circle if all the poles are inside the unit circle.

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Discrete-Time LTI Filtering Discrete-Time LTI Filtering ARMA, MA and AR Filters ARMA, MA and AR Filters Other commonly used terminology for the ﬁlters described include: Other commonly used terminology for the ﬁlters described include:

I Autoregressive moving average (ARMA) ﬁlter: I Moving average (MA) ﬁlter:

N M M X X X y(n) = − ak y(n − k) + bk x(n − k) y(n) = bk x(n − k) k=1 k=0 k=0 PM b z −k M H(z) = k=0 k X −k PN −k H(z) = bk z 1 + ak z k=1 k=0 has both poles and zeros I I has zeros only; no poles; is BIBO stable IIR I I FIR

Dr. Deepa Kundur (University of Toronto) Discrete-Time LTI Systems and Analysis 59 / 61 Dr. Deepa Kundur (University of Toronto) Discrete-Time LTI Systems and Analysis 60 / 61 Discrete-Time LTI Filtering ARMA, MA and AR Filters Other commonly used terminology for the ﬁlters described include:

I Autoregressive (AR) ﬁlter:

N X y(n) = − ak y(n − k) k=1 1 H(z) = PN −k 1 + k=1 ak z

I has poles only; no zeros

I IIR

Dr. Deepa Kundur (University of Toronto) Discrete-Time LTI Systems and Analysis 61 / 61