SIGNALS AND SYSTEMS

Aydın Akan

Department of Electrical and Electronics EngineeringUniversity of Istanbul

Avcilar, Istanbul 34850 TURKEYE-mail: akan@istanbul.edu.tr

k

c

......

k

0 12

-1-2

October 2001, Istanbul

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TABLE OF CONTENTS

1. CONTINUOUS AND DISCRETE TIME SIGNALS ANDSYSTEMS

1.1. INTRODUCTION TO SIGNALS

1.2. BASIC TRANSFORMATIONS ON SIGNALS

1.3. BASIC CONTINUOUS-TIME SIGNALS

1.4. BASIC DISCRETE-TIME SIGNALS

1.5. PERIODICITY OF DISCRETE-TIME SINUSOIDS

1.6. INTRODUCTION TO SYSTEMS

1.7. PROPERTIES OF SYSTEMS

2. LINEAR TIME-INVARIANT SYSTEMS

2.1. CONTINUOUS-TIME LTI SYSTEMS

2.2. DISCRETE-TIME LTI SYSTEMS

2.3. PROPERTIES OF LTI SYSTEMS

2.4. REPRESENTATION OF LTI SYSTEMS BY DIFFERENTIALEQUATIONS

2.5. REPRESENTATION OF LTI SYSTEMS BY BLOCK DIAGRAMS

3. CONTINUOUS-TIME FOURIER ANALYSIS

3.1. RESPONSE OF LTI SYSTEMS TO COMPLEX SINUSOIDS

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3.2. CONTINUOUS-TIME FOURIER SERIES

3.3. CONTINUOUS-TIME FOURIER TRANSFORM FOR APERI-ODIC SIGNALS

3.4. PROPERTIES OF THE CONTINUOUS-TIME FOURIER TRANS-FORM

3.5. SAMPLING OF CONTINUOUS TIME SIGNALS

4. FILTERING

4.1. IDEAL FILTERS

4.2. NON-IDEAL FREQUENCY SELECTIVE FILTERS

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1. CONTINUOUS AND DISCRETE TIME SIG-NALS AND SYSTEMS

1.1. INTRODUCTION TO SIGNALS

A signal is the outcome of a physical system. Signals are representedmathematically as functions of one or more independent variables. A speechsignal is represented by acoustic pressure as a function of time. A picture isrepresented as brightness function of two spatial (x and y) variables.

There are two basic types of signals:

– Continuous-time Signals

– Discrete-time Signals

In the case of continuous-time signals, the independent variable is contin-uous; these signals are defined for a continuum of values of the independentvariable. On the other hand, for discrete-time signals, the independent vari-able takes only a discrete-set of values. Hence discrete-time signals are onlydefined at discrete time instants.

A speech signal as a function of time (pressure or electrical voltage as afunction of time), [see Fig. 1] atmospheric pressure as a function of attitude,electrocardiogram ECG (or any other biological) signals as a function of timeare some examples of continuous-time signals.

The amount of precipitation in kg per month, the height of a child mea-sured and recorded every year are examples of discrete-time signals. Figures2 and 3 show examples of such signals.

t, Continuous-time variable x(t), Continuous-time signaln, Discrete-time variable x(n), Discrete-time signal

x(n) is defined only for integer values of n. For some discrete-time sig-nals, the independent variable is inherently discrete (e.g. precipitation permonth, height vs. year).

On the other hand, some discrete-time signals may represent samples ofa continuous-time signal. For example, to process a continuous time signalon a digital computer, we use samples of this signal at discrete-time signals.This is called “sampling of a continuous-time signal.”

4

0 2 4 6 8 10 12 14 16 18 20−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Time, t [msec]

Am

plitu

de, x

(t)

[V]

A speech segment

Figure 1: Example of a continuous-time signal: a 20 msec. speech segment.

x(t) −→ x(nT ) = x(n) n = ...− 1, 0, 1, ...

where T is called the sampling period. Finally, based on the nature of thetime and the amplitude variables, we have the following signal classification:

Amplitude / Time Continuous DiscreteContinuous ANALOG DISCRETE-TIME

Discrete QUANTIZED DIGITAL

1.2. BASIC TRANSFORMATIONS ON SIGNALS

a) Reversing in Time:Given a continuous-time signal x(t) (or a discrete-time signal x(n)), the

time reversed version of it, xr(t) (or xr(n)) is obtained by

xr(t) = x(−t)xr(n) = x(−n)

5

months

P(kg)

Jan

Feb

Mar

Apr

Nov

Dec

Figure 2: Example of discrete-time signal: the amount of precipitation inkg per month.

years

H(cm)

0 1 2 3 4

50

6070

80

90

. . .

. . .

Figure 3: Another example of discrete-time signal: the height of a childmeasured and recorded every year.

Figures 4 and 5 show a continuous-time signal x(t), and its time reversedversion, xr(t). A similar example can be given for a discrete-time signal.

b) Scaling in Time :A continuous-time signal x(t) (or a discrete-time signal x(n)), can be

scaled in time such that xs(t) (or xs(n)) is defined as

xs(t) = x(αt); α > 0xs(n) = x(dn); d > 1, integer

6

t

x(t)

Figure 4: A continuous-time signal x(t).

t

x(t)

Figure 5: xr(t), Time-reversed version of x(t).

Here, the real number α > 0 is called the “scaling factor”. When we haveα < 1, then x(αt) is an expanded version of x(t) in time. On the otherhand if α > 1, then x(αt) is shrunken (contracted) version of x(t). Fordiscrete-time signals the scaling factor d must be an integer value becausethe signal is only defined for integer values of n, and (dn) must also be aninteger.

x(αt) =

α < 1 expansionα = 1 x(t)α > 1 contraction

In Figs. 6, 7, and 8, we give a continuous-time signal x(t) and its scaledversions. Fig. 7 show x(2t), contracted signal, and Fig. 8 show x(0.5t),expanded signal.

c) Shifting in Time:

7

t

x(t)

Figure 6: A continuous-time signal x(t).

t

x(2 t)

Figure 7: Scaled signal x(2t); contraction.

Given a continuous-time signal x(t) (or a discrete-time signal x(n)), itstime-shifted or translated version xt(t) (or xt(n)) is obtained as

xt(t) = x(t− t0)xt(n) = x(t− n0)

where t0 and n0 indicate the shift amount. When this shift amount ispositive, shifting in time is called “delay”, and when it is negative shiftingis called “advance”. We have a continuous-time signal x(t) in. Fig 9 and itstime delayed version xt(t) = x(t− t0) in Fig. 10 by a shift amount t0 > 0.

t0 > 0 −→ Shift to the right −→ Delaying the signalt0 < 0 −→ Shift to the left −→ Advancing the signal

d) Scaling by a Constant:

8

t

x(t/2)

Figure 8: Scaled signal x(0.5t); expansion.

t

x(t)

x(0)

Figure 9: A given continuous-time signal x(t).

Given a signal x(t) or x(n), we can scale its magnitude by multiplyingthe signal by a constant c, such that

xc(t) = cx(t)xc(n) = cx(n)

e) Even and Odd Decomposition:Any real-valued signal can be decomposed into two parts: one “even

symmetric” and one “odd symmetric” component. That is,

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

where xe(t) indicates the even symmetric part and xo(t) indicates the oddsymmetric part of the signal. A signal is called even symmetric if it isidentical to its reflection about zero (or time-reversed version):

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t

x(t-t )

x(0)

t0

0

Figure 10: Time delayed signal xt(t) = x(t− t0).

xe(−t) = xe(t), ∀txe(−n) = xe(n), ∀n

A signal is referred to as an odd symmetric signal if

xo(−t) = −xo(t), ∀t

xo(−n) = −xo(n), ∀nFor an odd symmetric signal around zero, xo(0) = 0 or xo(0) = 0.

To obtain the even and odd components of a given signal,

x(t) = xe(t) + xo(t) (1)x(−t) = xe(−t) + xo(−t)x(−t) = xe(t)− xo(t) (2)

Adding both sides of equations (1) and (2), we get

x(t) + x(−t) = 2xe(t)

Hence, we obtain

xe(t) =x(t) + x(−t)

2.

Subtracting (2) from (1), we have

x(t)− x(−t) = 2xo(t)

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Then, we get

xo(t) =x(t)− x(−t)

2.

f) Periodicity of Signals:

A signal is periodic with period T if there exist a positive real numberT for which;

x(t) = x(t + mT ) for any integerm

The smallest possible value of T is called the fundamental period T0, andx(t) = x(t + T0). A constant signal is periodic with any choice of T (thereis no smallest possible value).

For discrete time signals,

x(n) = x(n + kN) for any integerk

A signal that is periodic with N is also periodic with integer multiplesof N , i.e., x(n) = x(n+2N) = x(n+3N) = .... A signal that is not periodicis called “aperiodic”.

1.3. BASIC CONTINUOUS-TIME SIGNALS

a) Unit-Step Function:The unit-step function is defined as

u(t) =

1 t > 00 t < 0

Figure 11 shows a unit-step function u(t). As we see, u(t) is not defined fort = 0 causing a discontinuity.

b) Unit-Impulse Function:The unit-impulse or delta dirac function is defined as the derivative of

the unit-step function:

11

t

u(t)

0

1

Figure 11: The unit-step function u(t).

δ(t) =d

dtu(t)

Conversely, u(t) can be obtained from δ(t) as a running integration;

u(t) =∫ t

−∞δ(τ)dτ

As we mention above, u(t) has a discontinuity at t = 0, hence we cannottake the derivative. So let’s define a non-ideal step function u∆(t), (see Fig.12) from which we can get the derivative;

δ∆(t) =d

dtu∆(t)

The δ∆(t) (shown in Fig. 13) can be viewed as a non-ideal impulse function.In fact, as ∆ → 0, δ∆(t) gets narrower and higher but maintains its unitarea. Finally,

lim∆→0

δ∆(t) = δ(t)

Figure 14 shows the ideal unit-impulse function in this as the limit case.

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t

u (t)

0

1

D

D

Figure 12: A non-ideal unit-step function u∆(t).

t

(t)Dd

0

1/D

D

Figure 13: Non-ideal impulse function, δ∆(t).

The value of δ(t) of t = 0 is actually ∞, thus the area under it is unity.However, we always assume that the amplitude value of the unit-impulsefunction is also unity.

∫ t

−∞kδ(τ)dτ = ku(t)

A very important property of impulse function is that, we can representany point of a signal by a scaled impulse, i.e.,

x(t)δ(t) = x(0)δ(t)

that is if we multiply a signal x(t) with a unit impulse function, we get justan impulse function at t = 0 scaled by the amplitude of the signal at thet = 0, i.e., x(0). Similarly if we multiply our signal x(t) with a shifted unitimpulse function δ(t − t0), we get another impulse function at t0 scaled bythe amplitude of the signal at t0, i.e., x(t0).

x(t)δ(t− t0) = x(t0)δ(t− t0).

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t

(t)d

0

1

Figure 14: The ideal unit-impulse function δ(t).

Later on, we will see how any signal can be represented as a weighted com-bination of shifted impulses.

c) Complex Exponentials/Sinuoids:A continuous-time complex exponential signal is given in general by

x(t) = Aeat

where the amplitude A and the phase a are complex in general. If A and aare real, we define real-valued exponential functions. When a > 0, we haveincreasing exponential (see Fig. 15), and when a < 0, we have decreasingexponential (see Fig. 16).

t

x(t)

A

Aeat

a>0

Figure 15: An increasing exponential signal.

If the phase a is complex, and pure imaginary, i.e., a = jΩ0; we have a“periodic signal” that is called a complex sinusoid;

x(t) = ejΩ0t

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t

x(t)

A

Aeat

a<0

Figure 16: A decreasing exponential signal.

where Ω0 is the radial (angular) frequency of the sinusoid. To see thatx(t) is periodic,

ejΩ0(t+T ) = ejΩ0t ejΩ0T

To satisfy the above equation, we need that

ejΩ0T = 1.

In fact, for Ω0 6= 0 there is a smallest possible T , T0 which is called thefundamental period of x(t), such that Ω0 T0 = 2π or

T0 = 2π/|Ω0|.

ThenejΩ0T0 = ej2π = 1

proving that x(t) is a periodic signal with period T0, or frequency Ω0.

d) Real sinusoids:A general real-valued sinusoidal signal is given by

x(t) = A cos(Ω0t + φ) (3)

where t is the time in seconds, Ω0 = 2πF0 is the fundamental radial fre-quency in radian/seconds, φ is the initial phase in radians, F0 is the fre-quency in Hertz. Fig. 17 shows a part of a (periodic) real-valued sinusoidalsignal with radial frequency Ω0, i.e., fundamental period T0 = 2π/Ω0.

Complex sinusoids and real sinusoids are related by the Euler identities:

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t

x(t)Acos( )W0 Ft +

AcosFT0

Figure 17: A continuous-time real-valued sinusoid.

ejΩ0t = cos(Ω0t) + j sin(Ω0t)e−jΩ0t = cos(Ω0t)− j sin(Ω0t)

cos(Ω0t) =ejΩ0t + e−jΩ0t

2

sin(Ω0t) =ejΩ0t − e−jΩ0t

2j

Moreover, we have that

A cos(Ω0t + φ) =A

2ejΩ0t ejφ +

A

2e−jΩ0t e−jφ

A cos(Ω0t + φ) = ReAej(Ω0t+φ)A sin(Ω0t + φ) = ImAej(Ω0t+φ)

1.4. BASIC DISCRETE-TIME SIGNALS

a) The-Unit Step Function:Discrete-time unit-step function u(n) is defined as 1 for positive n values

and 0 for negative n, i.e.,

u(n) =

1 n ≥ 00 n < 0

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Fig. 18 show a discrete-time unit-step function u(n). Notice that in discrete-time unit-step function, there is no ambiguity at n = 0, namely u(0) = 1.

n

u(n)

1

Figure 18: Discrete-time unit-step function u(n).

b) Unit-Impulse (Sample) Function:The discrete-time unit-impulse or unit-sample function δ(n) given in Fig.

19 is defined as 1 for n = 0 and 0 elsewhere, i.e.,

δ(n) =

1 n = 00 n 6= 0

n

d

1

(n)

. . .. . .

Figure 19: Discrete-time unit-impulse function δ(n).

As in the continuous-time case, discrete-time unit-step and unit-impulsefunctions are also related to each other. For example, unit-impulse functioncan be obtained from unit-step function by a “first order difference”, i.e.,

δ(n) = u(n)− u(n− 1)

Similarly, unit-step function can be represented as a combination of shiftedunit-impulse functions, i.e.,

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u(n) =∞∑

k=0

δ(n− k)

Furthermore, we can obtain the values of unit-step function for any n, as arunning-sum of a single unit-impulse function, i.e.,

u(n) =n∑

m=−∞δ(m)

This idea is illustrated in Figs. 20-(a) and 20-(b) for n < 0 and n ≥ 0respectively.

m

(m)d

u(n)=0

n<0

1

(a)

m

(m)d u(n)=1

n>0

1

(b)

Figure 20: Representation of unit-step as running-sum of unit-impulse, (a)for n < 0, (b) for n ≥ 0.

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c) Complex Exponentials/Sinusoids:In the discrete-time, we define complex exponentials as

x(n) = Aean = Aαn

In general, A and α are complex values. When A and α are real, we havethe following situations:

• |α| > 1: increasing exponential

• |α| < 1: decreasing exponential

• α > 0: all positive values

• α < 0: alternating signs

In Fig. 21, we see an increasing exponential with all positive values, α > 1,but in Fig. 22, we have a decreasing exponential with all positive values,0 < α < 1.

n

x(n)

A

A na>1a

......

Figure 21: An increasing exponential with positive values, α > 1.

Fig. 23, shows an increasing exponential with positive and negativevalues, α < −1, and Fig. 24, shows a decreasing exponential with positiveand negative values, −1 < α < 0.

If the exponent a is complex, and pure imaginary, i.e., a = jω0; we havea discrete-time complex sinusoid;

x(n) = ejω0n

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n

x(n)

A

A na <1a0<

......

Figure 22: A decreasing exponential with positive values, 0 < α < 1.

n

x(n)

A

A na<-1a

......

Figure 23: An increasing exponential with positive and negative values,α < −1.

where ω0 is the discrete angular frequency of the sinusoid. We willdiscuss the periodicity of discrete-time (complex and real valued) sinusoidsin detail.

c) Real Sinusoids:A discrete-time real-valued sinusoidal signal is given by

x(n) = A cos(ω0n + φ) (4)

where n is the integer sample index, ω0 = 2πf0 is the fundamental radial(or angular) frequency in radians, φ is the initial phase in radians, f0 is the

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n

x(n)

A

A na <0a-1<

......

Figure 24: A decreasing exponential with positive and negative values, −1 <α < 0.

normalized discrete frequency with no units.Complex- and real-valued sinusoids are related to each other in the same

way with their continuous-time counterparts:

Aej(ω0n+φ) = A cos(ω0n + φ) + jA sin(ω0n + φ)Ae−j(ω0n+φ) = A cos(ω0n + φ)− jA sin(ω0n + φ)

A cos(ω0n + φ) =A

2ej(ω0n+φ) +

A

2e−j(ω0n+φ)

A sin(ω0n + φ) =A

2jej(ω0n+φ) − A

2je−j(ω0n+φ)

1.5. PERIODICITY OF DISCRETE-TIME SINUSOIDS

Here we discuss two periodicity issues of discrete-time sinusoids.

1. Periodicity of the discrete frequency:As opposed to the continuous-time sinusoids, the frequency of discrete-

time sinusoids is not increased as we make ω0 larger. Consider a complexsinusoid, ejω0n. Now let’s add 2π to its frequency, i.e.,

ej(ω0+2π)n = ej2πn ejω0n = ejω0n

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since ej2πn = 1. We see that increasing the frequency of this signal by anamount of 2π, the signal remains the same, i.e., frequency ω0 +2π is equiva-lent to ω0. Hence we can say that the frequency of discrete-time sinusoids is periodic by multiples of 2π.Therefore, we only consider the frequency range 0 ≤ ω ≤ 2π or equivalently,−π ≤ ω ≤ π. The signals are slow at frequencies around 0 or 2π and fastaround ±π.

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1. Periodicity of the sinusoidin time:Now we consider the time periodicity of the sinusoid. For any signal to

be periodic, we need that the signal repeat itself after N samples, i.e.,

ejω0(n+N) = ejω0nejω0N

For the above to be equal to the original signal ejω0n, we require that

ejω0N = 1.

That is only possible ifω0N = k2π

i.e., integer multiples of 2π, (k integer). Hence, we have that

2π

ω0=

N

k.

It is also clear that the period of a discrete-time signal must be an integervalue. Therefore, for a discrete-time sinusoid to be periodic the ratio 2π

ω0

must be a rational value. The smallest possible integer N0 that satisfies thiscondition is called the fundamental period of the sinusoid.Example 1: Let us determine whether or not the signal

x(n) = ej2.3πn

is periodic, and if so, its fundamental period. As we see, the radial frequencyof the signal is ω = 2.3π rad. Considering mod2π of this value, we getfundamental frequency as ω0 = 0.3π rad. Now, the ratio

2π

ω0=

2π

0.3π=

203

=N

k

Now definitely, the value 203 = 6.6666... samples cannot be the period of a

discrete-time signal. However, it is a rational number, then x(n) is peri-odic. Now we can find an integer k such that k 20

3 is an integer. Hence fork = 3, the fundamental period N0 = 20 samples is calculated.Example 2: In this example we consider a real sinusoid,

x(n) = cos(0.5n).

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It looks like the frequency of this signal is ω0 = 0.5 rad. Calculating theratio

2π

ω0=

2π

0.5= 4π =

N

k.

4π is not an integer value and it cannot be the period of this signal. More-over, it is not a rational number. It is not possible to find an integerk such that k 4π is an integer. Hence the signal x(n) = cos(0.5n) is notperiodic.

1.6. INTRODUCTION TO SYSTEMS

A system performs a transformation on the input signal and generatesan output.

y(n) = T [x(n)]

In Fig. 25, we show both continuous-time and discrete-time systems. Theinput signal is denoted by x(t) (or by x(n) for discrete-time systems) andthe output by y(t) (or y(n)).

S Sx(t) y(t) x(n) y(n)

Figure 25: Continuous- and discrete-time systems

Systems can be used in groups to form larger systems, i.e., two sys-tems can be connected one after the other (serial or cascade) or in parallel.Figs. 26 and 27 show cascade and parallel operation of two systems S∞ andsystems S∈.

S S1 2x(t) y(t)

Figure 26: Serial or cascade connection of two systems.

1.7. PROPERTIES OF SYSTEMS

1. Systems with memory:

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S

S

1

2

x(t) y(t)+

Figure 27: Parallel connection of two systems.

Any system with transformation

y(t) = kx(t)

for a constant k is called memoryless system. For example, a resistor,v(t) = Ri(t) is memoryless. y(t) = x(t) that is called “identity system” isalso memoryless. Any other system must have memory to keep past valuesof input and output to generate the present output. For instance,

y(n) =x(n) + x(n− 1) + x(n− 2)

3is a system with memory as it stores and uses two previous input samples.

y(t) = 1/C

∫ t

−∞x(τ)d(τ)

where x(t) is the input current and y(t) is the output voltage of a capacitorthat is also a system with memory.

2. Invertible Systems:A system is invertible if by observing the output, we can determine its

input. The transformation must be 1-1 for us to get the systems inverse.Given the system equation

y(t) = 2x(t)

the inverse system can be determined easily as x(t) = 1/2 y(t).Example: Consider an accumulator system

y(n) =n∑

k=−∞x(k).

Let’s obtain its inverse transformation T−1:

y(n) = · · ·+ x(n− 2) + x(n− 1) + x(n) (5)y(n− 1) = · · ·+ x(n− 2) + x(n− 1) (6)

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Subtracting (6) from (5), we get the inverse transformation as

x(n) = y(n)− y(n− 1).

Hence the inverse of the accumulator system is a first order difference system(just like the inverse of integrator is a differentiator).

3. Superposition (Linearity):Given a system with input/ouput relation y(t) = T [x(t)], we have two

conditions for the system to be “linear”1) Homogeneity: If the system is applied a scaled version of the input,that is ax(t), and the output is

T [ax(t)] = aT [x(t)] = ay(t)

then system is called “homogeneous”.2) Additivity: Given a system with input/output pairs y1(t) = T [x1(t)]and y2(t) = T [x2(t)], and the system is applied the sum of the two inputsignals x1(t) + x2(t), and the output is

T [x1(t) + x2(t)] = T [x1(t)] + T [x2(t)] = y1(t) + y2(t)

then the system is called “additive”. Combining conditions (1) and (2), wehave the superposition or linearity property of systems. In general, if asystem is given combination of inputs

∑

k

akxk(t) = a1x1(t) + a2x2(t) + · · ·

, with T [xk(t)] = yk(t) and

T

[∑

k

akxk(t)

]=

∑

k

akT [xk(t)] =∑

k

akyk(t)

then this system is called linear.

4. Time Invariance:

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For a system with input/output relation y(t) = T [x(t)], if the system isapplied a time-shifted version of the input, i.e., x(t− t0), and the responseof the systems is

T [x(t− t0)] = y(t− t0)

then the system is called time-invariant. This means that delaying theinput causes just the same amount of delay at the output. The systemtreats the input signal in the same way, i.e., the system does not change orevolve with time. For instance,

y(t) = sin[x(t)], and y(n) = x(n) + 2x(n− 1)

systems are time-invariant (TI). However,

y(n) = nx(n− 1)

system is not time-invariant or it is time varying, due to the time-dependenceof the system equation.

5. Causality:Causality is the property of real-life systems. It means that the system

does not require any input or output value from the future times. A systemis causal if the output is a function of only present and previous values ofthe input. For example,

y(t) = x(t− 1), and y(n) = x(n)− 0.8x(n− 1)

systems are causal. However,

y(n) = 3x(n)− x(n + 1), and y(t) = x(t + 1)

systems are not causal or noncausal.Causal systems can be implemented physically and operated real-time.

However, non-causal systems can be realized to work only off-line usingpreviously stored information about past and future.

6. Stability:Stability is a property that all physical systems are required to have. It

means that given a non-diverging (or stable) input to a system, the out-put generated by the system must also be finite, or bounded. This concept

27

of stability is called Bounded-Input-Bounded-Output Stability. Hence anybounded input to a stable system does not cause the output to diverge.Then, a system is called Bounded-Input-Bounded-Output (BIBO)stable if a bounded input generates a bounded output. For example,

y(n) = x(n) + 0.8y(n− 1) and y(n) = x(n)− 0.8x(n− 1)

systems are stable, i.e., any bounded input will generate a bounded outputsignal. However,

y(n) = x(n) + 2y(n− 1)

system is unstable because of the 2 gain factor in the feedback from y(n−1)to y(n) which causes the output to blow up as n grows.

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2. LINEAR TIME-INVARIANT SYSTEMS

An important class of systems is Linear Time-Invariant Systems (LTI)systems. This class of systems satisfy the Linearity and Time-Invarianceproperties. Many physical process can be modelled by LTI systems and theyare easy to analyze. Recall that linearity implies that an infinite combinationof signals

x(t) =∞∑

k=1

akxk(t)

= a1x1(t) + a2x2(t) + · · ·generates the output

y(t) = a1y1(t) + a2y2(t) + · · ·

=∞∑

k=1

akyk(t)

where T [xk(t)] = yk(t).Before we go in detail about LTI systems we present the idea of signal

representation in terms of shifted impulses. Any single point of a signal canbe represented by a shifted and weighted impulse function. That is, if wemultiply a signal x(t) with a unit-step function δ(t), we get

x(t)δ(t) = x(0)δ(t)

which is a weighted impulse with x(0) at t = 0. As shown in Fig. 28, thesignal component at t = 0 can be represented by an impulse at t = 0 (δ(t))with an amplitude equal to the value of the signal at t = 0 (x(0)).

Now considering every single point of the signal and combining them,we can represent the whole signal by a combination of shifted and properlyweighted impulse functions, i.e.,

x(t) =∫ ∞

−∞x(τ) δ(t− τ) dτ

=∫ ∞

−∞x(t) δ(t− τ)dτ (7)

29

t

x(t)

x(0)d(t)

t

x(t)

x(0)d(t)=

d(t)

Figure 28: Representation of a continuous-time signal by impulses.

This is called the shifting property of impulse functions. An analogousrepresentation can be obtained for a discrete-time signal x(n) as given inFig. 29. For example, the value of x(n) at n = 1, or x(1) can be representedby a multiplication of the signal with a unit impulse shifted to x(1),

x(n)δ(n− 1) = x(1)δ(n− 1).

Repeating this for all values of n, we get

x(n) = · · ·+ x(−1)δ(n + 1) + x(0)δ(n) + x(1)δ(n− 1) + · · ·

=∞∑

k=−∞x(k)δ(n− k) (8)

n

x(n)

x(1)

n

d

=

d(n-1)

x(1) (n-1)

x(n)

1 1

Figure 29: Representation of a discrete-time signal by impulses.

Example 1: Consider a four sample discrete-time signal

x(n) =12

δ(n + 1) + δ(n)− 12

δ(n− 1) +14

δ(n− 2)

30

given in Fig. 30. This signal can be represented as a sum of four shiftedimpulses that are scaled by the amplitude of the samples, i.e.,

x(n) = x(−1) δ(n + 1) + x(0)δ(n) + x(1) δ(n− 1) + x(2) δ(n− 2)

We show all of these four components of the signal in Fig. 30.

2.1. CONTINUOUS-TIME LTI SYSTEMS

In this section we discuss how the output of an LTI system is related to itsinput. In the previous chapter, we express systems using a transformationform the input to the output. In the time domain continuous-time sys-tems input/output relation is usually represented by differential equations.Another relation we consider in this section is called “The ConvolutionIntegral”. In the following, we develop this very useful tool for the analysisof continuous-time LTI systems.

According to the shifting property of impulses, any signal can be repre-sented as a weighted combination of shifted impulses:

x(t) =∫ ∞

−∞x(τ)δ(t− τ)dτ (9)

Let h(t) be the response of the system given in Fig. 31, to the input δ(t)(unit-impulse function), i.e.,

T [δ(t)] = h(t)

Since the system is known to be time-invariant, the response of the systemto a shifted impulse should be

T [δ(t− τ)] = h(t− τ) (Time-Invariance)

By homogeneity condition of linearity, the response of this system to ashifted impulse, weighted by a scalar, x(τ)δ(t− τ) is,

T [x(τ)δ(t− τ)] = x(τ)T [δ(t− τ)]= x(τ)h(t− τ) (Homogeneity)

31

n

x(n)

10-1-2-3 2 3

1

1/2

-1/2

1/4

n

x(n)

10-1-2-3 2 3

1/2

d(n+1)

n1

0-1-2-3 2 3

x(n)d(n-1)

-1/2

n10-1-2-3 2 3

1

x(n)d(n)

n10-1-2-3 2 3

1/4

x(n)d(n-2)

=

+Figure 30: A discrete-time signal and its impulse representation.

32

S Sh(t) h(t- )(t)d (t- )d t t

Figure 31: Impulse response of a continuous-time system.

Finally by the additivity property of linear systems, the output of thesystem to combination of x(τ)δ(t− τ), ∀τ (or x(t)) should be the superpo-sition of the outputs to individual inputs x(τ)δ(t− τ),

∫ ∞

−∞x(τ) δ(t− τ) dτ = x(t) (10)

∫ ∞

−∞x(τ) h(t− τ) dτ = y(t) (Additivity) (11)

where y(t) is the output of this system to the input signal x(t). This relationis called The Convolution Integral and it is denoted by ’∗’,

y(t) = x(t) ∗ h(t) =∫ ∞

−∞x(τ) h(t− τ) dτ (12)

The impulse response h(t) is the signature of an LTI system. As such,an LTI system is fully characterized by its impulse response. For any giveninput signal, we can analytically calculate the corresponding output. More-over, all properties of the system are encrypted and hidden in the impulseresponse.

Properties of convolution integral:

1. Commutative: x(t) ∗ h(t) = h(t) ∗ x(t) The convolution operation iscommutative, therefore we have

y(t) = x(t) ∗ h(t) =∫ ∞

−∞x(τ) h(t− τ) dτ

= h(t) ∗ x(t) =∫ ∞

−∞x(t− τ) h(τ) dτ

which can be proven by a simple change of variables.

2. Associative:

x(t) ∗ [h1(t) ∗ h2(t)] = [x(t) ∗ h1(t)] ∗ h2(t)

This property is applied to cascade connection of two LTI systems.

33

3. Distributive:

x(t) ∗ (h1(t) + h2(t)) = (x(t) ∗ h1(t)) + (x(t) ∗ h2(t))

which is applied to parallel connection of LTI systems.

Example 2:The input signal to an LTI system is x(t) = e−atu(t) and the impulse

response of the system is h(t) = u(t) as shown in Fig. 32. Let’s calculatethe response of the system to x(t) by using convolution;

y(t) =∫ ∞

τ=−∞x(τ) h(t− τ) dτ

t

x(t)

t

h(t)

1

1

0 < a < 1

Figure 32: The input signal and impulse response of an LTI system.

We will solve this problem step by step, so we need x(τ), h(τ), h(−τ),and h(t − τ), then the product x(τ)h(t − τ) ∀t, τ . Now, let’s draw thesesignals,

34

1. x(τ) = e−aτu(τ)

2. h(τ) = u(τ), h(−τ) = u(−τ)

3. h(t− τ) = u(t− τ)

4.

x(τ)h(t− τ =

0 t < 0e−aτ u(t− τ) t ≥ 0

5. Then y(t) = 0, t < 0. For t ≥ 0, we only need to integrate the productobtain in (4).

y(t) =∫ ∞

−∞x(τ)h(t− τ)dτ (13)

=∫ ∞

0e−aτu(t− τ)dτ (14)

=∫ t

0e−aτdτ (15)

=−1a

e−aτ∣∣∣t0 (16)

=1− e−at

a(17)

Therefore, the total response of the system to x(t) is given by

y(t) =1− e−at

au(t), ∀t

and shown in Fig. 34.

2.2. DISCRETE-TIME LTI SYSTEMS

In this section we repeat the above derivations for discrete-time systemsto get their input/output relation. Discrete-time LTI systems are in generalrepresented the time domain by constant coefficient difference equations.We present here another relation between the input and the output whichis called “The Convolution Sum”. for the analysis of discrete-time LTIsystems.

35

By the shifting property of discrete-time unit impulse, any signal can berepresented as a weighted combination of shifted impulses:

x(n) =∞∑

k=−∞x(k)δ(n− k)

The output can be calculated as a linear combination of the responses toinput components x(k)δ(n−k) which are shifted and weighted impulses. Leth(n) denote the response to δ(n) then the response to δ(n− k) is h(n− k)due to the time-invariance property of the system:

T [δ(n− n)] = h(n− k) (Time-Invariance)

Also the response of this system to a shifted and weighted impulse, x(k)δ(n−k) is,

T [x(k)δ(n− k)] = x(k)T [δ(n− k)]= x(k)h(n− k) (Homogeneity)

Then the output of the system to the sum of x(k)δ(n − k), ∀k (or x(n))should be the sum of the outputs to individual inputs x(k)δ(n − k) by theadditivity property of linear systems:

∞∑

k=−∞x(k) δ(n− k) = x(n) (18)

∞∑

k=−∞x(k) h(n− k) = y(n) (Additivity) (19)

where y(n) is the response of this system to the input signal x(n). Thisrelation is called The Convolution Sum for discrete-time systems and itis again denoted by ’∗’,

y(n) = x(n) ∗ h(n) =∞∑

k=−∞x(k) h(n− k) (20)

h(n) is called as the impulse response of a discrete-time LTI system. Asbefore, a discrete-time LTI system is fully characterized by its impulse re-sponse.

Properties of convolution sum:

36

The discrete-time convolution has the following properties:

1. Commutative: x(n) ∗ h(n) = h(n) ∗ x(n), i.e., we can interchange therole of the input signal and the impulse response:

y(n) = x(n) ∗ h(n) =∞∑

k=−∞x(k) h(n− k)

= h(n) ∗ x(n) =∞∑

k=−∞x(n− k) h(k)

2. Associative:

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

This property is illustrated by the serial (cascade) connection of twoLTI systems shown in Fig. 35.

3. Distributive:

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

The distributive property can be applied to the parallel connection oftwo LTI systems shown in Fig. 36.

Example 3:We are given a discrete-time system with input

x(n) = αnu(n), 0 < α < 1

and the impulse responseh(n) = u(n)

given in Fig. 37To solve this problem step by step, we need x(k), h(k), h(−k), h(n− k),

and the product x(k)h(n− k). Now, let’s give and draw all these signals,

1. x(k) = αku(k)

2. h(k) = u(k), h(−k) = u(−k)

37

3. h(n− k) = u(n− k)

4.

x(k)h(n− k) =

0 n < 0αk u(n− k) n ≥ 0

5. Then we see that y(n) = 0, t < 0, and for n ≥ 0, we need to performthe sum of x(k)h(n− k) over k.

y(n) =∞∑

k=−∞x(k)h(n− k) (21)

=∞∑

k=0

αk u(n− k) (22)

=n∑

k=0

αk (23)

=1− αn+1

1− α, n ≥ 0 (24)

Therefore, the total output of this system to x(n) is given by

y(n) =

(1− αn+1

1− α

)u(n) ∀n

and shown in Fig. 39.Remark: Infinite and finite geometric series expansion identities for any0 < |α| < 1 are given by

∞∑

k=0

αk =1

1− α

N−1∑

k=0

αk =1− αN

1− α

2.3. PROPERTIES OF LTI SYSTEMS

An LTI system is fully characterized by its impulse response throughthe convolution integral or sum. Hence, the properties of the system can beinvestigated on the impulse response.

38

y(t) =∫ ∞

−∞x(τ)h(t− τ)dτ

=∫ ∞

−∞x(t− τ)h(τ)dτ

= x(t) ∗ h(t)

and

y(n) =∞∑

k=−∞x(k)h(n− k)

=∞∑

k=−∞h(k)x(n− k)

= x(n) ∗ h(n)

1. LTI Systems with/without memory:A system is memoryless if the output at any time depends only on the

input at that same time. For these systems the impulse response must be

h(t) = 0, t 6= 0.

For instance, y(t) = Kx(t) scaling system has impulse response h(t) = Kδ(t)(and for K = 1, it is called an identity system) which is a system withoutmemory. Same is true for discrete-time system y(n) = Kx(n), and h(n) =Kδ(n).

If a system has an impulse response h(t) or h(n) that is not zero for t 6= 0then that system has memory.

y(n) = x(n− 1) + x(n + 1)

is a system memory.Recall that a memoryless system with K = 1 is the identity system.

y(t) = x(t) ∗ δ(t) =∫ ∞

−∞x(τ)δ(t− τ)dτ = x(t)

y(n) = x(n) ∗ δ(n) =∞∑

k=−∞x(k)δ(n− k) = x(n)

39

Above equations are true because of the shifting property of the impulsefunction.

2. Invertibility of LTI systems:An LTI system is invertible if given the impulse response of the system,

we can obtain an inverse system h′(t), such that

h(t) ∗ h′(t) = δ(t)

which means that the equivalent of cascade of h(t) and h′(t) is an identitysystem as given in Fig. 40

x(t) ∗ h(t) = y(t)y(t) ∗ h′(t) = x(t) ∗ h(t) ∗ h′(t)

= x(t)

then we needh(t) ∗ h′(t) = δ(t).

Example 4:The delay system is defined as y(t) = x(t− t0), h(t) = δ(t− t0).

If t0 > 0, input signal is shifted to the right :delayIf t0 < 0, input signal is shifted to the left :advance

x(t− t0) = x(t) ∗ δ(t− t0)

Remark: The convolution of a signal with an impulse, is the same signal.The convolution of a signal with an shifted impulse shifts the signal to theposition of the impulse.

The inverse of the above system is;

h′(t) = δ(t + t0)y(t) ∗ h′(t) = x(t− t0) ∗ δ(t + t0) = x(t)h(t) ∗ h′(t) = δ(t− t0) ∗ δ(t + t0) = δ(t)

Then, the inverse of the delay system with h(t) = δ(t − t0) is the advancesystem with h′(t) = δ(t + t0), and

x(t) = y(t) ∗ δ(t + t0) = y(t + t0)

40

Example 5:The system with impulse response h(n) = u(n) is called summer or

accumulator.

y(n) =∞∑

k=−∞x(k)u(n− k) =

n∑

k=−∞x(k)

This system is invertible.

h′(n) = δ(n)− δ(n− 1)x(n) = y(n)− y(n− 1)

h(n) ∗ h′(n) = u(n) ∗ [δ(n)− δ(n− 1)]= u(n)− u(n− 1) = δ(n)

3. Causality of LTI systems:In general the output of a causal system depends only on the present

and past values of the input. Special to LTI systems, causality requires that

h(n) = 0 n < 0

y(n) =n∑

k=−∞x(k)h(n− k) =

∞∑

k=0

h(k)x(n− k)

For example h(n) = u(n) and its inverse h′(n) = δ(n)−δ(n−1) are casual.However, h(t) = δ(t− t0) for t0 < 0 is a non-casual system (advance).

4. Stability of LTI systems:A system is stable if every bounded input produces a bounded output.

Then for LTI systems, we have

|x(n)| < B ∀n

41

The output is

|y(n)| =

∣∣∣∣∣∣

∞∑

k=−∞x(n− k)h(k)

∣∣∣∣∣∣

|y(n)| ≤∞∑

k=−∞|x(n− k)||h(k)|

|y(n)| ≤ B∞∑

k=−∞|h(k)| ∀n

which is to say that if the impulse response is absolutely summable;

∞∑

k=−∞|h(k)| < ∞

then y(n) is bounded and the system is stable. The condition for continuous-time LTI systems is ∫ ∞

−∞|h(t)|dt < ∞

Delay systems h(n) = δ(n− n0) or h(t) = δ(t− t0) are stable

∞∑

−∞|h(n)| =

∞∑

−∞|δ(n− n0)| = 1 < ∞

∫ ∞

−∞|h(t)|dt =

∫ ∞

−∞|δ(t− t0)|dt = 1 < ∞

However, accumulator or integrator systems (h(n) = u(n), h(t) = u(t)) areunstable;

∞∑

−∞|u(n)| = ∞

∫ ∞

−∞|u(t)|dt = ∞

for the integrator system the output is

y(t) =∫ ∞

−∞x(τ)u(t− τ)dτ =

∫ t

−∞x(τ)dτ

5. Unit Step Response of LTI systems:

42

The response of a LTI system to a unit-step function is called the “UnitStep Response” and denoted by s(t) or s(n)

x(t) = u(t)

s(t) =∫ ∞

−∞x(t− τ)h(τ)dτ

=∫ ∞

−∞u(t− τ)h(τ)dτ

=∫ t

−∞h(τ)dτ

which is the area under h(t) up to time t. Hence the inverse would be

h(t) =d

dts(t) = s′(t)

that is the impulse response is the first derivative of the step response.Similarly of r discrete-time LTI systems

x(n) = u(n) (25)s(n) = u(n) ∗ h(n) (26)

=∞∑

k=−∞u(n− k)h(k) (27)

=n∑

k=−∞h(k) (28)

which means that step response is the sum of the values of h(n) up totime n. Then the inverse is a first order difference of the step responseh(n) = s(n)− s(n− 1).

2.4. REPRESENTATION OF LTI SYSTEMS BY DIFFER-ENTIAL EQUATIONS

a) Continuous-Time LTI systems:Linear, constant-coefficient differential equations are usually employed

to represent or model continuous-time LTI systems. Input and output of acontinuous-time system are related by a simple differential equation:

d

dty(t) + 2y(t) = x(t)

43

To find the output as a function of the input, we need to solve the aboveimplicit equation and get explicit representation

y(t) = yp(t) + yh(t)

yh(t) is the homogeneous solution and is determined by the auxiliary condi-tions. It does not depend on the input. For example, y(0) = y0. For such asystem to be linear, the output must be zero, when input is zero thereforeyh(t) must be zero which requires that the auxiliary conditions be

y(0) = y0 = 0.

If y0 6= 0 the system is not linear. Generally, a system can be representedby a linear part (that has zero auxiliary conditions and yh(t) = 0) plus theresponse to nonzero auxiliary conditions as illustrated by Fig. 41. yp(t) iscalled the private solution due to the input.

Causality of LTI systems described by differential equations depends onthe auxiliary condition as well. “Initial rest” condition specifies that if theinput x(t) = 0 for t ≤ t0, y(t) also is zero for t ≤ t0. However choosinga fixed point for auxiliary condition such as y(0) = 0 leads to non-causalsystems.

For a causal system initial rest is chosen so that y(t) = 0 when x(t) = 0but initial rest does not specify the auxiliary condition at a fixed point intime. A a summary, if we make the initial rest assumption, and if x(t) = 0for t ≤ t0, we need to solve y(t) for t > t0 using the condition y(t0) = 0which is called initial condition. Initial rest also implies time-invariance.x(t) → y(t), y(t0) = 0 x(t − T ) → y(t − T ), y(t0 + T ) = 0 (initialcondition).

A general N th-order linear constant-coefficient differential equation isgiven by

N∑

k=0

akdk

dtky(t) =

M∑

k=0

bkdk

dtkx(t) (29)

if N = 0 explicitly

y(t) =1a0

M∑

k=0

bkdk

dtkx(t)

44

a) Discrete-Time LTI systems: Linear constant-coefficient difference equa-tions are used to represent discrete-time LTI systems:

N∑

k=1

aky(n− k) =M∑

k=1

bkx(n− k) (30)

The system is incrementally linear, and it is linear if the auxiliary conditionsare zero.

y(n) =1a0

M∑

k=0

bkx(n− k)−N∑

k=1

aky(n− k)

with auxiliary conditions: y(−N), y(−N + 1)....y(−1)The above is a recursive system using a feedback from the output to the

input. Such systems have impulse response with infinite length (or time-support) and they are called “Infinite Impulse Response (IIR)” systems.However for N = 0

y(n) =1a0

M∑

k=0

bkx(n− k)

is a non-recursive system without a feedback. The output depends onlyinput values, but not the previous output values. This type of systems havefinite length impulse responses and called “Finite Impulse Response (FIR)”systems. The impulse response samples of FIR systems are basically thecoefficients of the difference equations, i.e.,

h(n) =

bna0

0 ≤ n ≤ M

0 otherwise

For example, h′(n) = δ(n)− δ(n−1) has finite nonzero points (length 2)and it is an FIR system. However h(n) =

(12

)nu(n) is the impulse response

(length ∞) of an IIR system.

2.5. REPRESENTATION OF LTI SYSTEMS BY BLOCKDIAGRAMS

The basic operations needed in the LTI systems are

45

• Addition of two signals

• Multiplication by a constant

• Delay

which are shown by the block symbols given in Fig. 42.

Example 6: Given a second order discrete-time system y(n) = b0x(n) +b1x(n−1)−a1y(n−1). We obtain its block diagram as follows (see Fig. 43:

w(n) = b0x(n) + b1x(n − 1) and y(n) = w(n) − a1y(n− 1) is called theDirect I implementation of the system (top figure). We can change the orderof the systems and obtain

x(n)− a1s(n− 1) = s(n)b0s(n) + b1s(n− 1) = y(n)

which is called the Direct II implementation of the same system (see bottomfigure). In general

y(n) =1a0

N∑

k=0

bkx(n− k)−N∑

k=1

aky(n− k)

we define

w(n) =N∑

k=0

bkx(n− k)

and

y(n) =1a0

w(n) +

N∑

k=1

(−ak)y(n− k)

for Direct I implementation and

s(n) =1a0

−

N∑

k=1

aks(n− k) + x(n)

and

y(n) =N∑

k=0

bks(n− k)

are used for Direct II implementation.

46

3. CONTINUOUS TIME FOURIER ANALYSIS

Representation of signals in terms of shifted and weighted impulses yieldsthe impulse response representation of LTI systems. Another importantrepresentation of LTI systems is the continuous and discrete-time Fouriertransforms and series. With this very useful representation, a signal will beexpressed as a combination of complex exponentials.

3.1. RESPONSE OF LTI SYSTEMS TO COMPLEX SINU-SOIDS

Consider a continuous-time LTI system with impulse response h(t), andassume this system is applied a complex exponential x(t) = est, then theoutput is

y(t) =∫ ∞

−∞h(τ)es(t−τ)dτ

= es(t)∫ ∞

−∞h(τ)e−sτdτ

= estH(s)

We conclude that the complex exponential est is an eigenfunction of LTIsystems where the transfer function (the Laplace transform of h(t)) H(s) isthe corresponding eigenvalue.

x(t) =∑

k

ak eskt

y(t) =∑

k

ak H(sk) eskt

Since complex exponentials are eigenfunctions of LTI systems, representinginput signals in terms of weighted exponentials makes it easy to obtain the

47

output and analyze the LTI systems.

3.2. CONTINUOUS-TIME FOURIER SERIES

Given a a continuous-time periodic signal with period T0

x(t) = x(t + T0) ∀t, 2π

T0= Ω0

Our aim is to find a series representation for this signal in term of complexsinusoids:

φk(t) = ejkΩ0t, k = 0,±1,±2, · · ·Let us use these harmonically related complex exponentials φk(t) to

represent the periodic signal x(t). The φk(t) are called the basis functionsand they all have fundamental frequency kΩ0 that is integer multiple of Ω0,(fundamental frequency of the signal). Thus they are periodic with T0

k thatmeans they are also periodic with T0. The Fourier basis φk(t) = ejkΩ0tform an orthogonal basis for the square summable functions space L2(R).Therefore, any signal with finite energy in a single period can be expressedby the following Fourier Series Expansion:

x(t) =∞∑

k=−∞ck ejkΩ0t (31)

For k = 0 we have c0 that represents the average value or DC term ofthe signal. For k = ±1 we have c1 and c−1, terms with the fundamentalfrequency of the original signal are called the first harmonics, c±1e

±jΩ0t withperiod T0. For |k| ≥ 2 we have other harmonics. k = ±N are referred to asthe N th harmonic component.

The representation of a periodic signal in this form is referred as theFourier series representation. If x(t) is real x∗(t) = x(t), then

x∗(t) = x(t) =∞∑

k=−∞c∗ke

−jkΩ0t (32)

=∞∑

k=−∞c∗−ke

jkΩ0t replaced k with − k (33)

(34)

48

Two representations of the same signal must be identical:

c∗−k = ck or c∗k = c−k

proving that the Fourier series coefficients ck are complex even symmetric.In order to determine the Fourier Series Coefficients for a given pe-

riodic signal x(t), let’s follow this derivation:The Fourier Series (FS) representation of the signal is

x(t) =∞∑

k=−∞cke

jkΩ0t

Multiply both sides with e−jnΩ0t and integrate over a period,

∫ T0

0x(t)e−jnΩ0tdt =

∫ T0

0

∞∑

k=−∞cke

jkΩ0t e−jnΩ0tdt

=∞∑

k=−∞ck

∫ T0

0ej(k−n)Ω0tdt (35)

Here, we use the orthogonality of the basis functions, i.e.,∫ T0

0ej(k−n)Ω0tdt =

∫ T0

0cos[(k − n)Ω0t]dt + j

∫ T0

0sin[(k − n)Ω0t]dt

=

T0 k = n0 k 6= n

= T0δ(n− k)

This tells us that φk(t) and φn(t) are orthogonal. Therefore, using this inequation (35), we get

∫ T0

0x(t)e−jnΩ0tdt = T0Cn

which gives us the FS expansion coefficients:

ck =1T0

∫

T0

x(t)e−jkΩ0tdt

x(t) =∞∑

k=−∞cke

jkΩ0t

49

The integration on t above is over a single period, so it could be from 0 toT0 as well as any other time period with T0 duration. ck are called FourierSeries Coefficients or spectral coefficients with c0 as the DC component.

c0 =1T0

∫

T0

x(t)d(t)

that is the average value of x(t). In general |ck|2, ∀k is plotted to give thespectral information of the signal. The value of |c1|2 is the energy of thefundamental harmonic at frequency Ω0, the value of |c2|2 is the energy ofthe second harmonic at frequency 2Ω0, etc.

Now, let’s explore the relation between the above complex FS expansionand the real or trigonometric FS representation. Recall that c∗k = c−k, and

x(t) = c0 +∞∑

k=1

[cke

jkΩ0t + c−ke−jkΩ0t

]

= c0 +∞∑

k=1

[cke

jkΩ0t + c∗ke−jkΩ0t

]]

= c0 +∞∑

k=1

2Re[cke

jkΩ0t]

Let ck = ak + jbk, then

x(t) = c0 + 2∞∑

k=1

Re(ak + jbk) [cos(kΩ0t) + j sin(kΩ0t)]

= c0 + 2∞∑

k=1

[ak cos(kΩ0t)− bk sin(kΩ0t)]

This is called the real or trigonometric Fourier series representation.

50

Remark: LTI System Interpretation.If x(t) is periodic with period T0 sec. and has a FS representation:

x(t) =∞∑

k=−∞cke

jkΩ0t

where Ω0 = 2π/T0 rad/sec., and it is applied to an LTI system with impulseresponse h(t) as shown in Fig. 44. The output of the system is

y(t) =∫ ∞

−∞h(τ)x(t− τ)dτ

=∫ ∞

−∞h(τ)

∞∑

k=−∞cke

jkΩ0(t−τ)dτ

=∞∑

k=−∞cke

jkΩ0t∫ ∞

−∞h(τ)e−jkΩ0τdτ

=∞∑

k=−∞ckH(kΩ0)ejkΩ0t

=∞∑

k=−∞dke

jkΩ0t

given as another FS representation with dk = ckH(kΩ0) as the FS coeffi-cients. Here we define the integral

H(Ω) =∫ ∞

−∞h(τ)e−jkΩτdτ

as the transfer function or frequency response of the above LTI system andH(kΩ0) is the response of the system to a sinusoid with frequency kΩ0.

Example 1: Obtain the FS representation for sinΩ0t. Without going intocalculation, we can use the following Euler identity,

sinΩ0t =ejΩ0t − e−jΩ0t

2j

x(t) = c−1e−jΩ0t + c1e

jΩ0t

From this we see the FS coefficients,

c−1 =−12j

, c1 =12j

, ck = 0 k 6= ±1

51

Example 2:Consider the following periodic signal shown in Fig. 45 with fundamental

period T0 sec. and fundamental angular frequency Ω0 = 2π/T0 rad/sec.

x() =

1 |t| < T1

0 T < |t| < T02

c0 =1T0

∫

T0

x(t)dt =1T0

∫ T1

−T1

1dt =2T1

T0

and for k 6= 0, we get the FS coefficients

ck =1T0

∫ T1

−T1

e−jkΩ0tdt =−1

jkΩ0T0e−jkΩ0t

∣∣∣T1−T1

=2

kΩ0T0

ejkΩ0T1 − e−jkΩ0T0

2j

=2

kΩ0T0sin kΩ0T1 =

sin kΩ0T1

kπ

and show them in Fig. 46.Any periodic signal can be approximated by a finite number of Fourier

Series coefficients.

xN (t) =N∑

k=−N

ckejkΩ0t ck =

1T0

∫

T0

x(t)e−jkΩ0tdt

As N →∞, the approximated signal xN (t) → x(t).

Remark: Drichlet ConditionsThe signal x(t) must be a continuous function and it must be square

integrable ∫

T0

|x(t)|2dt < ∞

(has finite energy over one period), then the coefficients ck are finite. x(t)has Fourier Series representation. For a periodic signal x(t) to have a FourierSeries Representation, it must satisfy the “Drichlet Conditions.”

52

1. Over one period, x(t) must be absolutely integrable∫

T0

|x(t)|dt < ∞ ⇒ |Ck| < ∞

2. During a single period, there are no more than ak finite number ofmaxima and minima.

3. In any finite interval of time, there are only a finite number of discon-tinuities that are also finite.

Periodic signal in Fig. 47 has discontinuities, but only 2 in one period, soit satisfies the Drichlet conditions. It has Fourier Series Representation andit can be approximated by a truncated Fourier Series expansion. However,it is very hard to approximate the edges of the signal and the error orimperfections at the edges of the square is named after “Gibbs.”

3.3. CONTINUOUS-TIME FOURIER TRANSFORM FORAPERIODIC SIGNALS

The Fourier Series expansion discussed in the previous section is veryuseful for the spectral analysis of continuous-time periodic signals. We alsoneed a similar tool for aperiodic signals which is the Fourier Transform(FT).

Given an aperiodic signal x(t), we can extend it to obtain a periodicversion x(t) as shown in Fig. 48 which can be expressed by a FS represen-tation,

(x)(t) =∞∑

k=−∞cke

jkΩ0t ck =1T0

∫ T02

−T02

x(t)e−jkΩ0tdt

ck =1T0

∫ T02

−T02

x(t)e−jkΩ0tdt =1T0

∫ ∞

−∞x(t)e−jkΩ0tdt

If we define

T0ck = X(Ω) =∫ ∞

−∞x(t)e−jΩtdt kΩ0 = Ω,

then we can write

x(t) =∞∑

k=−∞

1T0

X(kΩ0)e−jkΩ0t =Ω0

2π

∞∑

k=−∞X(kΩ0)ejkΩ0t

53

As T0 goes to ∞, then x(t) → x(t) and Ω0 → dΩ, and therefore

x(t) =12π

∫ ∞

−∞X(Ω)ejΩtdΩ The inverse Fourier Transform

X(Ω) =∫ ∞

−∞x(t)e−jΩtdt The Fourier Transform

Example 3:We calculate the Fourier Transform of a two-sided exponential signal

given in Fig. 49.

x(t) = e−a|t|, a > 0

The Fourier transform is

X(Ω) =∫ ∞

−∞e−a|t|e−jΩtdt

=∫ 0

−∞e(a−jΩ)tdt +

∫ ∞

0e−(a+jΩ)tdt

=1

a− jΩ+

1a + jΩ

=2a

a2 + Ω2

The Fourier spectrum of x(t) is shown in Fig. 50.

Example 4: Delta function, x(t) = δ(t). The Fourier Transform is

X(Ω) =∫ ∞

−∞δ(t)e−jΩtdt = 1

Example 5: Pulse (or rectangular gate) signal.

x(t) =

1 |t| < T1

0 |t| > T1

The Fourier Transform of this pulse is

X(Ω) =∫ T1

−T1

e−jΩtdt

= 2sin(ΩT1)

Ω= 2T1sinc(ΩT1)

54

where “sinc” function is defined as sin(x)/x. The FT of x(t) is given in Fig.??.

Example 6: Signal with pulse (or rectangular gate) spectrum shown in Fig.52.

1

W-W W

W X( )

Figure 52: A rectangular spectrum.

X(Ω) =

1 |Ω| < W0 |Ω| > W

The inverse Fourier Transform of X(Ω),

x(t) =12π

∫ W

−WejΩtdΩ =

sin(Wt)πt

=W

πsinc(Wt)

The signal x(t) is a sinc function in this case. Examples 5 and 6, point tothe duality property of the FT.

Remark: Fourier Transform of Periodic SignalsWe can obtain the Fourier transform of a periodic signal via the Fourier

series representation of it. We explain this using the following examples:

Example 7: Unit impulse spectrum

X(Ω) = 2πδ(Ω)

The inverse Fourier Transform of X(Ω) is,

x(t) =12π

∫ ∞

−∞2π δ(Ω)ejΩtdΩ = ej0t = 1

Now consider the FT that is a shifted impulse to frequency Ω0 shown in Fig.??.

X(Ω) = 2πδ(Ω− Ω0)

The signal that corresponds to X(Ω) is,

x(t) =12π

∫ ∞

−∞2π δ(Ω− Ω0)ejΩtdΩ = ejΩ0t

that is a complex sinusoid at frequency Ω0, which is clearly a periodic signal.

Example 8: Combination of complex sinusoids, i.e., Fourier Series.

55

The periodic signal ejΩ0t has a FT 2π δ(Ω − Ω0). So, we can expressany periodic signal as a combination of ejkΩ0t (that is the FS expansion)then take its FT which will be the combination of 2π δ(Ω− kΩ0). Given aperiodic signal,

x(t) =∞∑

k=−∞ck ejkΩ0t

then the FT is obtained by

X(Ω) =∞∑

k=−∞2π ck δ(Ω− kΩ0)

which is again impulses in frequency (separated by Ω0) and weighted by ck

(see Fig. 54).

2 c

W

W X( )

0

p

W 0W20W0W2- -

......

0

2 cp 12 cp 1

Figure 54: Fourier transform of a periodic signal.

Example 9: Sinus function.

x(t) = sin Ω0t =12j

[ejΩ0t − e−jΩ0t

]

ck = 0, k 6= ±1, then the FT of sinus function is

X(Ω) =12j

[2π δ(Ω− Ω0)− 2π δ(Ω + Ω0)]

We show the Fourier transform in Fig. ??.

Example 10: Cosine function.

x(t) = cos Ω0t =12

[ejΩ0t + e−jΩ0t

]

ck = 0, k 6= ±1, then the FT of cosine

X(Ω) =12

[2π δ(Ω− Ω0) + 2π δ(Ω + Ω0)]

which is given in Fig. 56.

W

W X( )

0W0W-

pp

Figure 56: Fourier transform of cosine function.

Example 11: A periodic impulse train.

56

Consider the periodic impulse train of period T sec. shown in Fig. ??.

x(t) =∞∑

k=−∞δ(t− kT )

The FS coefficients of this signal,

ck =1T

∫ T/2

−T/2δ(t)e−jkΩ0tdt =

1T

and using the result of Example 8, we get the FT of the pulse train

X(Ω) =∞∑

k=−∞2π ck δ(Ω− kΩ0) =

2π

T

∞∑

k=−∞δ(Ω− k

2π

T)

that is another impulse train in frequency (of period 2π/T ) illustrated inFig. 58.

X( )

......-

W

W

p2 / T

p2 / Tp2 / T

Figure 58: FT of the periodic impulse train.

3.4. PROPERTIES OF THE CONTINUOUS-TIME FOURIERTRANSFORM

1. Linearity: The FT is a linear operation. x1(t) → X1(Ω) andx2(t) → X2(Ω) then

a1x1(t) + a2x2(t) → a1X1(Ω) + a2X2(Ω)

2. Symmetry: If x(t) is real then its FT is conjugate symmetric, X(−Ω) =X∗(Ω). For example,

x(t) = e−atu(t) → X(Ω) =1

a + jΩ

andX(−Ω) =

1a− jΩ

= X∗(Ω)

If the signal x(t) is real, and that

X(Ω) = ReX(Ω)+ jImX(Ω)X(Ω) = |X(Ω)|ej 6 X(Ω)

then the symmetry property becomes,

57

• The real part of the FT ReX(Ω) is even symmetric

• The imaginary part of the FT ImX(Ω) is odd symmetric

• The magnitude of the FT |X(Ω)| is even symmetric

• The phase of the FT 6 X(Ω) is odd symmetric

Furthermore,

• If x(t) is real and even, then X(Ω) is also real and even; X(−Ω) =X(Ω)

• If x(t) is real and odd, then X(Ω) is pure imaginary and odd; X(−Ω) =−X(Ω)

• Even and odd symmetric parts of the signal

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

corresponds to

X(Ω) = ReX(Ω)+ jImX(Ω)then xe(t) → ReX(Ω) and xo(t) → ImX(Ω).

3. Time Shifting: If x(t) has a FT X(Ω), then the time shifted version ofx(t),

x(t− t0) → e−jΩt0X(Ω) = |X(Ω)|ej(6 X(Ω)−Ωt0)

that is shifting the signal in time just adds a phase to its FT, the magnituderemains the same.

4. Differentiation and Integration: The derivative of x(t) has the FT

d

dtx(t) → 1

2π

∫ ∞

−∞jΩX(Ω)ejΩtdΩ

x′(t) → jΩX(Ω)

and the integration of x(t) has the FT∫ t

−∞x(τ)dτ → 1

jΩX(Ω) + πX(0)δ(Ω)

58

5. Time and Frequency Scaling: If a signal x(t) has the FT X(Ω), thenits time-scaled version would have

x(at) →∫ ∞

−∞x(at)e−jΩtdt =

1|a|x(τ)e−j(Ω/a)τdτ =

1|a|X

(Ωa

)

6. Duality: In general

g(t) → f(Ω)−−f(t) → 2πg(−Ω)

Dual of differentiation in time (property 4) is

−jtx(t) → d

dΩX(Ω)

the dual of time shifting (property 3) is modulation, i.e.,

ejΩ0tx(t) → X(Ω− Ω0)

and the dual of integration in time (property 4) is

−1jt

x(t) + πx(0)δ(t) →∫ ∞

−∞X(η)dη

Rectangular pulse signal and sinc signal are Duals of each other. Letx1(t) be a pulse

x1t =

1 |t| < T1

0 |t| > T1

then its FT is a sinc function,

X1(Ω) =2 sin(ΩT1)

Ω= 2T1sinc

ΩT1

π

Further let x2(t) be a sinc signal:

x2(t) =sin(Wt)

πt=

W

πsinc

(Wt

π

)

59

then its FT is a frequency pulse,

X2(Ω) =

1 |Ω| < W0 |Ω| > W

which is the result of Duality property of the FT.

7. Parseval’s Relation: This is the property which indicates that the FTpreserves the energy of the signal, i.e., the energy of the signal calculated inthe time-domain or in the frequency domain are identical.

x(t) → X(Ω)

∫ ∞

−∞|x(t)|2dt =

12π

∫ ∞

−∞|X(Ω)|2dΩ = Ex

A similar energy relation is valid for periodic signals,

1T0

∫

T0

|x(t)|2dt =∞∑

k=−∞|ck|2

is the average energy of the signal in one period.

8. Convolution Property: This property is actually very useful for thefrequency domain analysis of LTI systems. We consider an LTI system withimpulse response h(t), and assume an input signal x(t) is applied to thissystem. The output is

y(t) = x(t) ∗ h(t) =∫ ∞

−∞x(τ) h(t− τ) dτ

Then the FT of the output,

Y (Ω) =∫ ∞

−∞

[∫ ∞

−∞x(τ) h(t− τ)dτ

]e−jΩtdt

=∫ ∞

−∞x(τ)

[∫ ∞

−∞h(t− τ)e−jΩtdt

]dτ

= H(Ω)∫ ∞

−∞x(τ)e−jΩτdτ

= H(Ω)X(Ω)

60

where H(Ω) is the FT of the impulse response and it is called the FrequencyResponse of the LTI system.

y(t) = h(t) ∗ x(t) → Y (Ω) = H(Ω)X(Ω)

Example 12: The delay system.

h(t) = δ(t− t0) → H(Ω) = e−jΩt0

Y (Ω) = H(Ω)X(Ω) = e−jΩt0X(Ω)

y(t) = x(t− t0)

Example 13: Differentiator.

y(t) =d

dtx(t)

Y (Ω) = jΩX(Ω) = H(Ω)X(Ω)

H(Ω) = jΩ

Example 14:Given an LTI system with h(t) = e−atu(t), a > 0 and input signal

x(t) = e−btu(t), b > 0, find the output using frequency methods.X(Ω) = 1/(b + jΩ) and H(Ω) = 1/(a + jΩ) and then the output is

Y (Ω) = 1/(b + jΩ)(a + jΩ).

Y (Ω) =1

b− a

[1

a + jΩ− 1

b + jΩ

]

y(t) =1

b− a

[eatu(t)− ebtu(t)

]

Example 15:Another LTI system with h(t) = e−tu(t) is given

x(t) =+3∑

k=−3

cke−jk2πt

61

c0 = 1, c1 = c−1 = 1/4, c2 = c−2 = 1/2, c3 = c−3 = 1/3

H(Ω) =1

1 + jΩ

X(Ω) =+3∑

k=−3

2πckδ(Ω− 2πk)

Y (Ω) = X(Ω)H(Ω) =+3∑

k=−3

2πckH(2πk)δ(Ω− 2πk)

=+3∑

k=−3

(2πck

1 + j2πk

)δ(Ω− 2πk)

y(t) =+3∑

k=−3

(ck

1 + j2πk

)ej2πkt

9. Modulation Property:Assume that two signals are multiplied such that

r(t) = s(t)p(t) ↔ R(Ω) =12π

[S(Ω) ∗ P (Ω)]

3.5. SAMPLING OF CONTINUOUS TIME SIGNALS

A continuous-time signal can be sampled or discretized by multiplyingit with an impulse train, p(t);

p(t) =∞∑

k=−∞δ(t− kT )

r(t) = s(t)p(t) =∞∑

k=−∞s(t)δ(t− kT ) =

∞∑

k=−∞s(kT )δ(t− kT )

The Fourier Transform of the impulse train,

P (Ω) =2π

T

∞∑

k=−∞δ(Ω− k

2π

T)

62

Then the FT of the sampled signal,

R(Ω) =12π

[S(Ω) ∗ P (Ω)] =1T

∞∑

k=−∞S(Ω) ∗ δ(Ω− k

2π

T)

R(Ω) =1T

∞∑

k=−∞S(Ω− k

2π

T)

The sampling of a continuous-time signal is illustrated in the time andfrequency domains in Fig. ??.

For a non-aliased sampling, i.e, for the spectra of S(Ω) not to overlap inR(Ω), we need that

2π

T− Ω1 ≥ Ω1 ⇒ 2π

T≥ 2Ω1

We call 2πT = Ωs as the sampling frequency.

2π

T= Ωs ⇒ Ωs ≥ 2Ω1

is called the Nyquist sampling criteria. The original signal can be recov-ered from r(t) by using a low-pass filter provided that the Nyquist Criteriais satisfied. We show in Fig. 60 the recovery of the original signal as acombination of sinc functions.

S( )W

WW1W1-

A

t

s(t)

Figure 60: The recovery of the continuous-time signal.

The frequency response of LTI systems is generally presented in loga-rithmic (dB) scale. These plots are called Bode Diagrams.

H(Ω) =Y (Ω)X(Ω)

Frequency response

|Y (Ω)| = |H(Ω)||X(Ω)||H(Ω)| : Magnitude response of the system.|Y (Ω)| : Magnitude spectrum of the output.|X(Ω)| : Magnitude spectrum of the input.

6 Y (Ω) = 6 H(Ω) + 6 X(Ω)

6 H(Ω) : Phase response of the system.6 Y (Ω) : Phase spectrum of the output.6 X(Ω) : Phase spectrum of the input.

63

The magnitude response in dB is defined as |H(Ω)|(dB) = 20log10|H(Ω)|which is then graphed as a function of Ω as shown in Fig. ??.

64

4. FILTERING

Frequency selective Filters are devices that allow some frequency com-ponents at the input signal to appear at the output, but eliminate otherfrequencies.

4.1. IDEAL FILTERS

1. Ideal Low-Pass Filter:

The magnitude response of an ideal low-pass filter is

|Hlp(Ω)| =

1 |Ω| < ΩC

0 |Ω| > ΩC

In Fig. 62 we give the magnitude and phase responses of an ideal low-pass filter.

W

W

|H ( )|

WcWc-

lp

PassbandStopband Stopband

W

W

H ( )lp =aW

1

Figure 62: Magnitude and phase responses of an ideal low-pass filter.The impulse response of a zero phase filter would be symmetric around

zero, but it is non-causal. So a realizable or causal filter will have an impulseresponse that is symmetric around α > 0 as displayed in Fig. ?? we give theimpulse response of a zero phase (but non-causal) filter, and its time-shifted(non-zero phase) version.

2. Ideal High-Pass Filter:

An ideal high-pass filter is the complementary of the ideal low-pass filter.The magnitude response is given by;

|Hhp(Ω)| =

1 |Ω| > ΩC

0 |Ω| < ΩC

and shown in Fig. 64.

W

W

|H ( )|

WcWc-

hp

Passband Stopband Passband

1

Figure 64: Magnitude response of an ideal high-pass filter.

65

3. Band-Pass Filter:

A band pass filter is the combination of a low-, and high-pass filters. Anideal band pass filter magnitude response given in Fig. ?? that is.

|Hbp(Ω)| =

1 Ω1 < |Ω| < Ω2

0 otherwise

4.2. NON-IDEAL FREQUENCY SELECTIVE FILTERS

Ideal filters are not possible to implement in practice, due to their sharptransitions and constant responses. However, we approximate the ideal fil-ters using some polynomial approximations which are called non-ideal filters.

• Butterworth

• Chebychev I and II

• Elliptic

66

x( )

h( )

1

1

t

t

t

t

h(- )

1

t

t

h(t- )

1

t

tt

t<0

1

t

t >=0

h(t- ) t

Figure 33: Calculation steps of a convolution integral using graphics.

67

t

y(t)

1/a

Figure 34: The response of the system to input x(t).

h = h1 * h2x y h1x y1 h2 y

h2x y2 h1 y=

Figure 35: Cascade connection of two discrete-time LTI systems.

h = h1 + h2x yh1 y1

h2 y2= +x y

Figure 36: Parallel connection of two discrete-time LTI systems.

68

n

x(n) = u(n)

1

0< <1a

n

h(n) =u(n)

1

. . .

an

Figure 37: The input signal and impulse response of a discrete-time LTIsystem.

69

k

1

0< <1a

. . .

x(k) = u(k)ak

k

h(-k)

1

k

h(k) =u(k)

1

k

h(n-k)

k

h(n-k)

n < 0

n >= 0

n

n

Figure 38: Calculation steps of a convolution sum.

70

n

y(n)

1/1-a

...

Figure 39: The response of the system to input x(n).

h(t)x(t) y(t) h'(t) x(t)

h(t) * h'(t) = (t)x(t) x(t)d

Figure 40: An LTI system and its inverse cascaded.

Linear System with zeroaux cond.

x(t)

y (t)h

+y (t)p

y(t)

Figure 41: An incrementally linear system model.

+

x (n)2

1x (n) x (n)1 x (n)2+

x(n) a a x(n)

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

x(t) xx-h

t

x( ) d( )t t

Figure 42: Block diagram symbols used to represent LTI systems.

71

+

z -1

+

z -1

x(n) w(n)b0

b1

y(n)

-a1

+

z -1

+

z -1

b0

b1

y(n)

-a1

x(n) s(n)

+

z -1

+ b0

b1

y(n)

-a1

x(n) 1

1

Figure 43: Block diagram of a second order discrete-time system.

h(t)x(t) y(t) = x(t) * h(t)

Figure 44: Periodic input to an LTI system.

t

x(t)

......1

T1-T1 T0-T0

Figure 45: A periodic pulse train.

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k

c

......

k

0 12

-1-2

Figure 46: Fourier series coefficients of the periodic pulse train.

t

x(t)

......1

T1-T1 T0-T0

Figure 47: Periodic pulse train with 2 discontinuities in a period.

t

x(t)

......T1-T1 T0-T0

~

t

x(t)

T1-T1

Figure 48: An aperiodic signal and its periodic extension.

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t

x(t)

1

0 < a

Figure 49: A two-sided exponential signal.

X( )

2/a

W

W

1/a

-a a

Figure 50: Fourier transform of two-sided exponential.

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