Discrete-Time Signals and Systems

Chapter 2: Discrete-Time Signals and Systems

Discrete-Time Signals and Systems


Reference: Sections 2.1 - 2.5 of

Dr. Deepa Kundur

John G. Proakis and Dimitris G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 4th edition, 2007.

University of Toronto

Dr. Deepa Kundur (University of Toronto)


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Elementary Discrete-Time Signals

Signal Symmetry

1. unit sample sequence (a.k.a. Kronecker delta function): 1, for n = 0 δ(n) = 0, for n 6= 0

Even signal:

x(−n) = x(n)

Odd signal: x(−n) = −x(n) 2. unit step signal: u(n) =

1, 0,

for n ≥ 0 for n < 0

ur (n) =

n, for n ≥ 0 0, for n < 0 -7

Note: δ(n)

x(n)

x(n)

2

2

2

1

1

1

x(n)

3. unit ramp signal:

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

n

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

n

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

= u(n) − u(n − 1) = ur (n + 1) − 2ur (n) + ur (n − 1)

u(n) = ur (n + 1) − ur (n) Dr. Deepa Kundur (University of Toronto)


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n



Signal Symmetry

Signal Symmetry 1

x(n)

(x(n)+x(-n))/2 1

0.5

1 2

Even signal component: xe (n) = [x(n) + x(−n)]

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

-0.5

Odd signal component: xo (n) = 12 [x(n) − x(−n)]

2 3 4

5 6

7 8

n

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

-0.5

-1

Note: x(n) = xe (n) + xo (n)

1

0.5 -6 -5 -4 -3 -2 -1 0 1

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n

7 8

2 3 4

5 6

7 8

n

-6 -5 -4 -3

-2 -1

-1


odd part

0.5

-1


5 6

(x(n)-x(-n))/2

1


2 3 4

-1

x(-n)

-0.5

even part

0.5

0 1

2 3 4

-0.5

5 6

n

7 8


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Simple Manipulation of Discrete-Time Signals

Simple Manipulation of Discrete-Time Signals I Find x(n) − x(n + 1).

I

Transformation of independent variable: I

time shift: n → n − k, k ∈ Z I

I

2

time scale: n → αn, α ∈ Z I

I

3

Question: what if k 6∈ Z? Question: what if α 6∈ Z?

I I

3 2

2

1

1

Additional, multiplication and scaling: I

x(n)

-3 -2 -1 0 1

amplitude scaling: y (n) = Ax(n), −∞ < n < ∞ sum: y (n) = x1 (n) + x2 (n), −∞ < n < ∞ product: y (n) = x1 (n)x2 (n), −∞ < n < ∞

-1

2 3 4

5 6

7 8 9 10


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20

n

-1

-2

-2

-3


15


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Simple Manipulation of Discrete-Time Signals II

Simple Manipulation of Discrete-Time Signals I Find x( 32 n + 1).

x(n)

3

2

1

1 -3 -2 -1 0 1

-1 -1 -2

2 3 4

-1

5 6

2

1

n < −1 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 > 12

x(n)-x(n+1) 1 15

7 8 9 10

20

n

-1

-1

-3 -x(n+1)



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3

2

-1

3n 2

0 if

3n 2

5 2

4 11 2

7

17 2

10 23 2

13 29 2

16 35 2

19 > 19

x( 3n + 1) 2 + 1 is an integer; undefined otherwise undefined x(1) = 1 undefined x(4) = 2 undefined x(7) = 3 undefined x(10) = 2 undefined x(13) = 1 undefined x(16) = −1 undefined x(19) = −2 + 1 is an integer; undefined otherwise


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Input-Output Description of Dst-Time Systems

+ 1). input/ excitation

3 2

2

1

1 -3 -2 -1 0 1

0 if


Simple Manipulation of Discrete-Time Signals Graph of

+1 < − 12 − 12 1



x( 23 n

3n 2

2 3 4

-2 -3


5 6

This signal is undefined for values of n that are not even integers and zero for even integers not shown on this sketch.

7 8 9

10

11

12

13 14

x(n)

Discrete-time System

I

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

I

“black box” representation:

-1 -2

output/ response

Discrete-time signal

Discrete-time signal

n

y(n)

T

x(n) −→ y (n) Discrete-Time Signals and Systems

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Classification of Discrete-Time Systems

Time-invariant vs. Time-variant Systems

Why is this so important? I

I

mathematical techniques developed to analyze systems are often contingent upon the general characteristics of the systems being considered for a system to possess a given property, the property must hold for every possible input to the system I I

I

I

T


for every input x(n) and every time shift k.

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Time-invariant vs. Time-variant Systems

Linear vs. Nonlinear Systems

Examples: time-invariant or not? I y (n) = A x(n) I y (n) = n x(n) p I y (n) = x(n) + x 2 (n − 2) I y (n) = x(−n) I y (n) = x(n + 1) 1 I y (n) = 1−x(n+2) I

T

x(n) −→ y (n) =⇒ x(n − k) −→ y (n − k)

to disprove a property, need a single counter-example to prove a property, need to prove for the general case


Time-invariant system: input-output characteristics do not change with time a system is time-invarariant iff

I I

Linear system: obeys superposition principle 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

y (n) = e 3x(n) Ans: Y, N, Y, N, Y, Y, Y



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Linear vs. Nonlinear Systems

Causal vs. Noncausal Systems

Examples: linear or not? I y (n) = A x(n) I y (n) = n x(n) p I y (n) = x(n) + x 2 (n − 2) I y (n) = x(−n) I y (n) = x(n + 1) 1 I y (n) = 1−x(n+2) I

y (n) = e

I

I

Causal system: output of system at any time n depends only on present and past inputs a system is causal iff y (n) = F [x(n), x(n − 1), x(n − 2), . . .] for all n

3x(n)

Ans: Y, Y, N, Y, Y, N, N



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Stable vs. Unstable Systems

Examples: causal or not? I y (n) = A x(n) I y (n) = n x(n) p I y (n) = x(n) + x 2 (n − 2) I y (n) = x(−n) I y (n) = x(n + 1) 1 I y (n) = 1−x(n+2) y (n) = e

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Causal vs. Noncausal Systems

I


I

I

Bounded Input-Bounded output (BIBO) Stable: every bounded input produces a bounded output a system is BIBO stable iff |x(n)| ≤ Mx < ∞ =⇒ |y (n)| ≤ My < ∞ for all n.

3x(n)

Ans: Y, Y, Y, N, N, N, Y



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Stable vs. Unstable Systems

The Convolution Sum

Examples: stable or not? I y (n) = A x(n) I y (n) = n x(n) p I y (n) = x(n) + x 2 (n − 2) I y (n) = x(−n) I y (n) = x(n + 1) 1 I y (n) = 1−x(n+2) I

Recall:

∞ X

x(n) =

x(k)δ(n − k)

k=−∞

y (n) = e 3x(n) Ans: Y, N, Y, Y, Y, N, Y



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The Convolution Sum

The Convolution Sum

Let the response of a linear time-invariant (LTI) system to the unit sample input δ(n) be h(n). Therefore,

T

δ(n) −→ h(n) T

δ(n − k) −→ h(n − k)

y (n) =

T

α δ(n − k) −→ α h(n − k)

∞ X

x(k)h(n − k) = x(n) ∗ h(n)

k=−∞

T

x(k) δ(n − k) −→ x(k) h(n − k) ∞ ∞ X X T x(k)δ(n − k) −→ x(k)h(n − k) k=−∞

for any LTI system.

k=−∞ T

x(n) −→ y (n)



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Properties of Convolution

Properties of Convolution

Associative and Commutative Laws:

Distributive Law:

x(n) ∗ h(n) = h(n) ∗ x(n) [x(n) ∗ h1 (n)] ∗ h2 (n) = x(n) ∗ [h1 (n) ∗ h2 (n)] x(n)

h1(n)

h2(n)

x(n) ∗ [h1 (n) + h2 (n)] = x(n) ∗ h1 (n) + x(n) ∗ h2 (n) h1(n) + h2(n)

y(n)

h1(n) x(n)

+

h1(n) * h2(n) = h2(n) * h1(n) x(n)

h2(n)


h1(n)

h2(n)

y(n)


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Finite impulse response (FIR):

h(k)x(n − k)

y (n) =

k=−∞

= =

−1 X

∞ X

h(k)x(n − k)

Infinite impulse response (IIR):

h(k)x(n − k)

k=0

y (n) =

h(k)x(n − k)

∞ X

h(k)x(n − k)

k=0

k=0

How would one realize these systems? Two classes: recursive and nonrecursive.

where h(n) = 0 for n < 0 to ensure causality.


M−1 X k=0

h(k)x(n − k) +

k=−∞ ∞ X

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Finite-Impulse Reponse vs. Infinite Impulse Response

For a causal system, y (n) only depends on present and past inputs values. Therefore, for a causal system, we have: y (n) =



Causality and Convolution

∞ X

y(n)


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Infinite Impulse Response System Realization


I

Consider an accumulator:

There is a practical and computationally efficient means of implementing a family of IIR systems that makes use of . . .

y (n) =

I


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=

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Linear Constant-Coefficient Difference Equations I

k=0 n−1 X




y (n) =

Memory requirements grow with increasing n!



n X

x(k) n = 0, 1, 2, . . .

k=0

. . . difference equations.


n X

x(k) +

Example for a linear constant-coefficient difference equation (LCCDE): y (n) = y (n − 1) + x(n) Initial conditions: at rest for n < 0; i.e., y (−1) = 0

x(k) + x(n) I

k=0

= y (n − 1) + x(n) ∴ y (n) = y (n − 1) + x(n)

General expression for Nth-order LCCDE: N X

recursive implementation

ak y (n − k) =

k=0

M X

bk x(n − k)

a0 , 1

k=0

Initial conditions: y (−1), y (−2), y (−3), . . . , y (−N)



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ak y (n − k) +

k=1

M X

bk x(n − k)

k=0

is equivalent to the cascade of the following systems:

...

bk x(n − k) nonrecursive | {z } k=1 input 1 N X y (n) = − ak y (n − k) + v (n) recursive |{z} |{z} k=1 output 2 input 2 v (n) = |{z} output 1

+

+

+

+

+

+

+

...

M X

+

+

...

y (n) = −

N X

Direct Form I IIR Filter Implementation

...

Direct Form I vs. Direct Form II Realizations

+

LTI All-zero system

LTI All-pole system

Requires: M + N + 1 multiplications, M + N additions, M + N memory locations Dr. Deepa Kundur (University of Toronto)


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Direct Form II IIR Filter Implementation +

Direct Form II IIR Filter Implementation Adder:

+ +

+

+

+

Unit delay:

+

+

+

Constant multiplier: +

+

+

Unit advance: +

+

+

+

...

+

+

... Signal multiplier:

...

...

+

...

...

+

...

+

...

+

LTI All-pole system

For N>M

LTI All-zero system

Requires: M + N + 1 multiplications, M + N additions, M + N memory locations Dr. Deepa Kundur (University of Toronto)

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Requires: M + N + 1 multiplications, M + N additions, max(M, N) memory locations Dr. Deepa Kundur (University of Toronto)


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Discrete-Time Signals and Systems

Discrete-Time Signals and Systems

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