ch1 signals systems

Signals and Systems

Basic Signals and Systems

Vijaya Krishna A

Vijaya Krishna A EC-253: Signals and Systems

Signals

Definition

A signal is a function of one or more independent variables.

Signals represent information.

A continuous signal is a function of one or more continuousvariables.

A discrete signal/sequence is a function of one or morediscrete variables.

In this course, we deal with functions of a single variable.

A continuous signal is typically taken to be a function of time.

Notation:

Continuous signal: x(t)Discrete signal: x [n]

x [n] is undefined for non-integer values of n.

n is unitless.


Signal Classification

Real/complex

Deterministic/random:

A deterministic signal is completely specified for all−∞ < t <∞ (−∞ < n <∞).A random signal is one that takes random values at any t (n).It is a signal about which there is uncertainty before its actualoccurrence.

Periodic/Aperiodic

Even/odd

Energy/power


Energy/Power

The energy of a signal x(t) is given by

E =

∫ ∞−∞|x(t)|2dt

Similarly

E =∞∑

n=−∞|x [n]|2

The power of a signal x(t) is given by

P = limT→∞

1

2T

∫ T

−T|x(t)|2dt

Similarly

P = limN→∞

1

2N + 1

N∑n=−N

|x [n]|2


Energy/Power

A signal is called an energy signal iff 0 < E <∞. In this case,P = 0.

A signal is called a power signal iff 0 < P <∞. In this case,E =∞.

For a periodic power signal with fundamental period T , wehave

P =1

T

∫ T

0|x(t)|2dt

If P =∞, then the signal is neither an energy signal nor apower signal.

Example: x(t) = t

Find the power/energy of the following:1 x(t) = e−atu(t), a > 0.2 x [n] = u[n]


Transformations of the Independent Variable

Shifting:

y(t) = x(t − t0): Shift x(t) to the right by t0 units.y(t) = x(t + t0): Shift x(t) to the left by t0 units.Similarly for x [n − n0] and x [n + n0].

Scaling: y(t) = x(at), where a is a non-zero real number.

|a| < 1: The signal is expanded.|a| > 1: The signal is compressed.

Reflection: y(t) = x(−t)

The signal is reflected about t = 0.Same as scaling with a = −1.

To find y(t) = x(at + b),

first shift by b,and then scale by a.


Periodic Signals

A continuous signal x(t) is said to be periodic if there exists apositive T such that

x(t) = x(t + T )

for all t.

The fundamental period T0 of x(t) is the smallest value of Tfor which the above equation holds.

If x(t) is a constant signal, its period is undefined.

A discrete signal x [n] is said to be periodic if there exists apositive N such that

x [n] = x [n + N]

for all n.

The fundamental period N0 is defined similarly.


Even and Odd Symmetry

A continuous signal x(t) is said to be even if

x(−t) = x(t)

A continuous signal x(t) is said to be odd if

x(−t) = −x(t)

The definitions are similar for discrete signals.

Properties:

If x(t) is odd, then x(0) = 0.If x(t) is even, then

∫ a

−ax(t)dt = 2

∫ a

0x(t)dt

If x(t) is odd, then∫ a

−ax(t)dt = 0

even × even = even.even × odd = odd.odd × odd = even.


Even-Odd Decomposition

Any arbitrary signal x(t) can be expressed uniquely as

x(t) = xe(t) + xo(t)

where the even part xe(t) and the odd part xo(t) are given by

xe(t) =1

2[x(t) + x(−t)]

xo(t) =1

2[x(t)− x(−t)]

Find the even and odd parts of x(t) = e−tu(t)


Unit Step and Unit Impulse

The discrete unit step is given by

u[n] =

{1 n≥0

0 n < 0

The discrete unit impulse is given by

δ[n] =

{1 n = 0

0 n 6=0

Properties:

δ[n] = u[n]− u[n − 1]

u[n] =∑∞

k=0 δ[n − k]

u[n] =∑n

k=−∞ δ[k]

x [n]δ[n − n0] = x [n0]δ[n − n0]: δ[n] is a sampling function.

x [n] =∑∞

k=−∞ x [k]δ[n − k]


The Continuous case

The unit step function is given by

u(t) =

{1 t > 0

0 t < 0

u(t) has a discontinuity at t = 0.The unit impulse (Dirac impulse) is a generalized functionLet

δε(t) =

{1ε 0 < t < ε

0 otherwise

The unit impulse is defined as

δ(t) = limε→0

δε(t)


Unit Impulse

δ(t) is a signal for which

δ(t) = 0 for t 6= 0

∫ ∞−∞

δ(t)dt = 1

Properties:

u(t) =

∫ t

−∞δ(τ)dτ or, u(t) =

∫ ∞0

δ(t − τ)dτ

x(t)δ(t − t0) = x(t0)δ(t − t0) : δ(t) is a sampling function.∫∞−∞ x(t)δ(t − t0)dt = x(t0)

x(t) =∫∞−∞ x(τ)δ(t − τ)dτ


Continuous Sinusoids

Considerx(t) = A cos(ω0t + φ)

where

A is the amplitude

ω0 is the fundamental frequency in radians/sec

ω0 = 2πf0, where f0 is the fundamental frequency incycles/sec or Hz

T0 = 2π|ω0| is the fundamental period

φ is the phase shift in radians

x(t) is periodic for any choice of ω0.

Sinusoids with different fundamental frequencies are distinct.

Let x0(t) = A cos(ω0t + φ0). x1(t) = A cos(ω1t + φ1) iscalled a harmonic of x0(t) if ω1 = kω0 for some integer k .


Exponentials

A real exponential is of the form

x(t) = Ceat

where C and a are real numbers.

When a is imaginary (and C = 1), we have x(t) = e jω0t

By Euler’s relation, we have

e jω0t = cosω0t + j sinω0t

e jω0t is periodic with a fundamental period of T0 = 2π|ω0| .

The definition of harmonics is similar.

A general complex exponential has growing/decaying sinusoidsas its real and imaginary parts.


Discrete Sinusoids

A discrete sinusoid is given by

x [n] = A cos(ω0n + φ)

whereω0k = 2π

N is the fundamental frequency in radians/sample,where k is an integer. N is the fundamental period.f0N = 1

N is the fundamental frequency in cycles/sample.

WKT,cosω0n = cos(ω0 + 2kπ)n

for any integer k .Hence, discrete sinusoids of fundamental frequencies ω0 andω0 + 2kπ (or, f0 and f0 + k) are identical.


Discrete Sinusoids

−8 −6 −4 −2 0 2 4 6 80

0.2

0.4

0.6

0.8

1

f0=0

−8 −6 −4 −2 0 2 4 6 8−1

−0.5

0

0.5

1

f0=1/8

−8 −6 −4 −2 0 2 4 6 8−1

−0.5

0

0.5

1

f0=1/4


Discrete Sinusoids

−8 −6 −4 −2 0 2 4 6 8−1

−0.5

0

0.5

1

f0=1/2

−8 −6 −4 −2 0 2 4 6 8−1

−0.5

0

0.5

1

f0=1

−8 −6 −4 −2 0 2 4 6 8−1

−0.5

0

0.5

1

f0=9/8


Discrete Sinusoids: Periodicity

Let the discrete sinusoid x [n] = cosω0n be periodic withperiod N (N > 0).

cosω0n = cosω0(n + N) = cos(ω0n + ω0N)

⇒ ω0N = 2πk .

⇒ ω0 = (2π)kN , or f0 = k

N .

∴ a discrete sinusoid is periodic if and only if f0 is rational.

Let the fundamental period be N. The fundamental frequencyis given by 2π

N = ω0k radians/sample.

Similar results hold for x [n] = e jω0n

Ex: Find the fundamental frequency of cos 3π4 n


Discrete Sinusoids

0 2 4 6 8 10 12 14 16 18 20−1

−0.5

0

0.5

1cos(πn/4)

0 2 4 6 8 10 12 14 16 18 20−1

−0.5

0

0.5

1cos(3πn/4)


Discrete Sinusoids

x [n] = cos 2πf0n can be regarded as a sampled version ofx(t) = cos 2πf0t, sampled at integer values of t.

In general, if we sample x(t) = cos 2πF0t at a samplingfrequency of Fs Hz (Ts = 1

Fssec), we get

x [n] ≡ x(nTs) = cos 2πF0nTs = cos 2πF0

Fsn = cos 2πf0n

Discrete frequency f0 is also called relative frequency andnormalized frequency.

We can relate f0 to F0 only if we know Fs .

A discrete sinusoid with a fundamental frequency of 2πN has

only N harmonics.


Systems

Definition

A system is an entity that operates on input signals to produceoutput signals.

A system can be a device, a program, or a natural system.

We consider systems with a single input and a single output.

Notation:

x(t)→ y(t)y(t) = H{x(t)}

We want to

Understand/Modify existing systems.Design new systems.

For these tasks, we need mathematical models for systems.

The models should be accurate enough.They should have enough structure to facilitate analysis.


System Properties

Memory:A systems is called memoryless if the output at any value of t (n)depends only on the input at the same value of t (n)

Otherwise, the system is said to have memory.

For systems with memory, the output at any time can dependon future inputs also.

Causality:A system is said to be causal if the present output does notdepend on future inputs.

A system is anticausal if the present output depends only onthe present and future inputs.

A system is noncausal if the present output depends on boththe past and the future inputs.

Memoryless systems are causal.


System Properties

Invertibility:A system is said to be invertible if distinct inputs lead to distinctoutputs. In such systems, the input can be recovered back fromthe output.Stability:A system is said to be stable if a bounded input leads to abounded output.For a stable system, if |x(t)| < Mx <∞, ∀t,then |y(t)| < My <∞, ∀tTime InvarianceA system is said to be time invariant if a a time shift in the inputsignal results in an identical time shift in the output signal.x(t)→ y(t) =⇒ x(t − t0)→ y(t − t0)


Linearity

A system is said to be linear if y1(t) = H{x1(t)} andy2(t) = H{x2(t)} implies

H{a1x1(t) + a2x2(t)} = a1y1(t) + a2y2(t)

This is known as the superposition property. It includes

Additivity: H{x1(t) + x2(t)} = y1(t) + y2(t)Homogeneity: H{a1x1(t)} = a1y1(t)

Examples:

Integration/differentiation.Linear constant coefficient ordinary differential/differenceequations.

Exercise: Find whether the following systems are linear:

y [n] = nx [n]y [n] = Re{x [n]}y(t) = 5x(t) + 2


Exercises

Find which of the system properties hold for the following systems:

y(t) =

{0, t < 0

x(t) + x(t − 2) t≥0

y(t) = Re{x(t)}y [n] = nx [n]


ch1 signals systems

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