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CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed

Lecture material courtesy of Dr. Anwar M. Mirza Lecture No. 02Date: June 11, 2012

1SIGNALS AND SYSTEMS

1.0. Introduction1.1. Continuous-Time and Discrete-Time Signals

1.1.1. Examples and Mathematical Representations1.1.2. Signal Power and Energy

1.2. Transformations of the Independent Variable1.2.1. Examples of the Transformations of the Independent Variable1.2.2. Periodic Signals1.2.3. Even and Odd Signals

1.3. Exponential and Sinusoidal Signals1.3.1. Continuous-Time Complex Exponential and Sinusoidal Signals1.3.2. Discrete-Tine Complex Exponential and Sinusoidal Signals1.3.3. Periodicity Properties of Discrete-Time Complex Exponentials

1.4. The Unit Impulse and Unit Step Functions1.4.1. The Discrete-Time Unit Impulse and Unit Step Sequences1.4.2. The Continuous-Time Unit Step and Unit Impulse Functions

1.5. Continuous-Time and Discrete-Time Systems1.5.1. Simple Examples of Systems1.5.2. Interconnections of Systems

1.6. Basic System Properties1.6.1. Systems with and without Memory1.6.2. Invertibility and Inverse Systems1.6.3. Causality1.6.4. Stability1.6.5. Time Invariance1.6.6. Linearity

1.7. Summary

Department of Computer Engineering

College of Computer & Information Sciences, King Saud UniversityAr Riyadh, Kingdom of Saudi Arabia



1.1.2 Signal Power and Energy

The total energy over the time internal t 1≤ t ≤ t2in a continuous-time signal x (t) is defined as

∫t1

t2

|x( t)|2dt (1.4)

where |x| denotes the magnitude of the (possibly complex) number x.

The time averaged power is given by

1(t2−t1 )∫t1

t2

|x( t)|2dt (1.4-1)

Similarly, the total energy in a discrete-time signalx [n] over the time interval n1≤n≤n2 is defined as

∑n=n1

n2

|x [ t ]|2 (1.5)

The average power over the interval in this case is given by

1n2−n1+1

∑n=n1

n2

|x [t ]|2 (1.5-1)

It is interesting to find out the power and energy in a signal over an infinite time interval, i.e., for −∞<t<+∞ or for −∞<n<+∞. In these cases the total energy and power carried out by the signal are given by:

For continuous-time (CT) case:

Total Energy E∞≜ limT→∞

∫−T

T

|x (t )|2dt=∫−∞

∞

|x (t)|2dt (1.6)

Total Average Power P∞≜ limT→∞

12T ∫

−T

T

|x (t)|2dt (1.8)

For discrete-time (DT) case:

Total Energy E∞≜ limN→∞

∑n=−N

+N

|x [n]|2= ∑n=−∞

+∞

|x [n ]|2 (1.6)

Total Average Power P∞≜ limN →∞

12N+1 ∑

n=−N

+N

|x [n]|2 (1.8)

Three important cases can be identified:





Case 1: Signals with finite total energy, i.e., E∞<∞:

Such a signal must have zero average power. For example, in continuous case, if E∞<∞, then

P∞=limT→∞

E∞

2T=0

An example of a finite-energy signal is a signal that takes on the value of 1 for 0≤ t ≤1 and 0 otherwise. In this case, E∞=1 and P∞=0.

Case 2: Signals with finite average power, i.e., P∞<∞:

For example, consider the constant signal where x [n]=4. This signal has infinite energy, as

E∞= limN →∞

∑n=−N

+N

|x [n]|2= limN→∞

∑n=−N

+N

42=…+16+16+16…

However, the total average power is finite,

P∞≜ limN →∞

12N+1 ∑

n=−N

+N

|x [n ]|2= limN→∞

12N+1 ∑

n=−N

+N

42

¿ limN →∞

162N+1 ∑

n=−N

+N

1

¿ limN →∞

16 (2N+1 )2N+1

¿ limN →∞

16=16

Case 3: Signals with neither E∞ nor P∞ finite:

A simple example of such a case could be x (t )= t. In this case both E∞and P∞are infinite.

1.2 Transformations of the Independent Variable

The transformation of a signal is one of the central concepts in the field of signals and systems.





We will focus on a very limited but important class of signal transformations that involves the modifications of the independent variable, i.e., the time axis.

1.2.1 Examples of Transformations of the Independent Variable

Time Shift

The original and the shifted signals are identical in shape, but are displaced or shifted along the time-axis with respect to each other. Signals could be termed as delayed or advanced in this case.

Signals Discrete-Time (DT) Continuous-Time (CT)

Original

Delayed

Advanced

Such signals arise in applications such as radar, sonar and seismic signal processing. Several receivers placed at different locations receive the time shifted signals due to the transmission time they take while passing through a medium (air, water or rock etc.).





Time Reversal (Reflection)

In this case, the original signal is reflected about the time = 0. For example, if the original signal is some audio recording, then the time reversed signal would be the audio recording played backward.

Signals Discrete-Time (DT) Continuous-Time (CT)

Original

Time Reversed



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Time Scaling

In this case, if the original signal is x (t), the time variable is multiplied with a constant to get a time-scaled signal, e.g., x (2 t), x (5 t ) ,or x (t /2). If we think of the signal x (t) as audio recording, then x (2 t) is the audio recording played at twice the speed and x (t /2) is the recording played at half of the speed.

Signals Continuous-Time (CT)

Original

Stretching

Compressing

General Case of the Transformation of the Independent Variable



Original Signal

Time Shifted Signal



A general case for the transformation of independent variable is the one in which for the original signal x (t) is changed to the form x (αt+β ), where α and β are given numbers. It has the following effects on the original signal:

The general shape of the signal is preserved. The signal is linearly stretched if |α|<1. The signal is linearly compressed if |α|>1. The signal is delayed (shifted in time) if β<0. The signal is advanced (shifted in time) if β>0. The signal is reversed in time (reflected) if α<0.

Example 1.1

Given the signal x (t) as shown in the figure below, the signal x (t+1 ) can be obtained by shifting the given signal to the left by one unit (as depicted in the following figure).

The original signal can be written mathematically as:

x (t )={ 012−t0

¿¿

¿

ifififif

¿¿

¿

t<00≤t<11≤ t<2t ≥2¿

¿



Original Signal

Time Reversed Signal



To determine x (t+1 ), the above definition of the original signal can be used. For example, to calculate the value of x (t+1 ) at t=0, we determine x (1 ), which according to the above definition is x (1 )=1, therefore, x (t+1 ) at t=0 is equal to 1.

Similarly, the signal x (−t+1 ) can be obtained using the mathematical definition or figure of the original signal x (t ). If we use the mathematical definition, then making the following table could be useful.

t −t+1 x (−t+1 )−2 3.0 0

−1.5 2.5 0−1 2.0 0

−0.5 1.5 0.50 1.0 10.5 0.5 11 0.0 11.5 −0.5 02 −1.0 02.5 −1.5 03 −2.0 0

Plotting of this signal would give us the following figure:

In MATLAB®, the original signal can be written as an inline function. This function can then be used to plot the original signal, the shifted signal and the time-reversed signal using the following MATLAB® code.





>> g = inline(' ((t>=0)&(t<1)) + (2-t).*((t>=1) & (t<2))','t');>> t = -3:0.001:3;>> subplot(3,1,1), plot(t,g(t)), axis([-3 3 -0.1 1.1]), title('Original Signal')>> subplot(3,1,2), plot(t,g(t+1)), axis([-3 3 -0.1 1.1]), title('Time-Shifted Signal')>> subplot(3,1,3),plot(t,g(-t+1)),axis([-3 3 -0.1 1.1]), title('Time-Reversed Signal')

-3 -2 -1 0 1 2 3

0

0.5

1

Original Signal

-3 -2 -1 0 1 2 3

0

0.5

1

Time-Shifted Signal

-3 -2 -1 0 1 2 3

0

0.5

1

Time-Reversed Signal





1.2.2 Periodic Signals

A periodic continuous-time signal x (t) is defined as

x (t )=x (t+T )

where T is a positive number called the period.

A typical example is that of a sinusoidal signal x (t )=sin (t) for −∞<t<+∞. This is shown in the figure below:

For the above signal, the period is T=2π . It can be noticed that for any time t :

sin (t+2π )=sin(t)

Also

sin ( t+m2π )=sin( t)

where m is an integer.

Periodic signals are defined analogously in discrete-time.

The fundamental period T 0 of x (t) is the smallest positive value of T for which the equation x (t )=x (t+T ) holds.

A discrete-time signal x [n] is periodic with period N , where N is a positive integer, if it is unchanged by a time-shift of N , i.e., if

x [n ]=x [n+N ]

for all values of n.

The fundamental period N 0 of x [n] is the smallest positive value of N for which the equation x [n]=x [n+N 0] holds.





An example of a discrete-time periodic signal is given in the following diagram.

1.2.3 Even and Odd Signals

A signal x (t) or x [n] is defined as an even signal if it is identical to its time-reversed counterpart, i.e., with its reflection about the origin.

Even continuous-time Signal x (−t )=x( t)Even Discrete-time Signal x [−n]=x [n]

A signal x (t) or x [n] is defined as an odd signal if,

Odd continuous-time Signal x (−t )=−x (t)Odd Discrete-time Signal x [−n ]=−x [n]

As a special case, the odd signal must be zero at t=0 or n=0. Examples of odd signals are given in the following figure.

x (t)

−6 π −4π −2π 0 2π 4 π 6 π t



A Discrete-Time Signal

Even Part Odd Part



An important fact is that any signal (continuous-time or discrete-time) can be broken into a sum of two signals, one of which is even and one of which is odd.

Signal Component Mathematical Form

Continuous-time Signalx (t)Even Part E v {x ( t ) }=12 [x ( t )+x (−t)]

Odd Part Od {x (t ) }=12 [ x (t )−x (−t)]

Discrete-time Signalx [n]Even Part E v {x [n]}=12 [x [n]+x [−n] ]

Odd Part Od {x [n]}=12 [x [n]−x [−n]]



Continuous-time Real Exponential with a>0

C

x(t)=Ceat, C>1, a>1

t

Continuous-time Real Exponential with a<0

C

x(t)=Ceat, C>0, a<0

t



1.3 Exponential and Sinusoidal Signals1.3.1 Continuous-Time Complex Exponential and Sinusoidal

Signals

A continuous-time complex signal x (t) can be written as

x (t )=C eat (1.20)where C and a are, in general, complex numbers.

Real Exponential Signals

In this case both C and a are real numbers, and x (t) is called a real exponential.

Periodic Complex Exponential and Sinusoidal Signals





Now we consider the case of complex exponentials where a is purely imaginary. More, specifically, we consider:

x (t )=e jω0 t (1.21)

An important property of this signal is that it is periodic. Let’s explore this.

By definition x (t) is periodic, if

x (t )=x (t+T )

for some real value of T . Therefore,

e jω0 t=e jω0 (t+T )

Or,

e jω0 t=e jω0 t e jω0T

⇒e jω0T=1 (1.23)

This equation can be true,

1 If, ω0=0, then x (t )=1, which is periodic for any value of T .2 If, ω0≠0, then the fundamental period T 0 of x (t ), i.e. the smallest value of T for

which equation (1.23) holds, is

T 0=2π|ω0|

(1.24)

Replacing the value of Twith this T 0 in equation (1.23), and using Euler’s formula, that is,

e jω0T=cos (ω0T )+ jsin (ω0T )We get

e jω0T=cos (2π )+ jsin (2 π )=1+ j0=1

Therefore, the signal x (t ) as given by equation (1.21) is a periodic signal. Similarly, it can be shown that the signal x (t )=e− jω0 t has the same fundamental period.

A closely associated signal is the sinusoidal signal

x (t )=A cos (ω0 t+ϕ) (1.25)



Continuous-Time Sinusoidal Signal

x(t)=A cos( 0 t+)

t

A

A cos()



Using Euler’s formula, the complex exponential of equation (1.21) can be written as

x (t )=e jω0 t=cos ( j ω0t )+ j sin ( j ω0 t) (1.26)

Similarly, the sinusoidal signal of equation (1.25) can be written as

A cos (ω0t+ϕ)=A ( e j (ω0t+ϕ)+e− j(ω0 t+ϕ )

2 )= A2e jϕe jω0 t+ A

2e− jϕ e− jω0 t

(1.27)

It can noted that the two exponentials in equation 1.27 have complex amplitudes. If we define Re {c } as the real component of a complex number c, and I m {c } as the imaginary part of c, then

A cos (ω0t+ϕ)=A R e {e j (ω0 t+ϕ)} (1.28)

A sin(ω0 t+ϕ)=A I m {e j(ω0 t+ ϕ)} (1.29)

Frequency and Period

From equation 1.24, it is clear that the fundamental period T 0 of a continuous-time sinusoidal or a periodic complex exponential signal, is inversely proportional to the |ω0|, which is called the fundamental frequency. If we decrease the value of the magnitude of ω0, we slow down the rate of oscillations and hence increase the period T 0. Alternatively, if we increase the value of the magnitude of ω0, we increase the rate of oscillations and hence decrease the period T 0.

Energy and Power

Over the one fundamental period T 0 of a continuous-time sinusoidal or a periodic complex exponential signal, the signal energy and power can be determined as:





Eperiod=∫0

T0

|e jω0 t|2dt=∫0

T 0

1dt=T0 (1.30)

Pperiod=1T0

∫0

T 0

|e jω0 t|2dt= 1T0

∫0

T 0

1dt=T 0T 0

=1 (1.31)

As there are an infinite number of periods as t ranges from −∞ to +∞, the total energy integrated over all time is infinite. The total average power is however remains 1, as by definition,

P∞=limT→∞

12T ∫

−T

T

|e jω0 t|2dt= limT →∞

12T2T=1 (1.32)

Harmonics of a Periodic Complex Exponential

We have noted that,

e jω T0=1 (1.33)

which implies that ωT 0 is a multiple of 2π , i.e.,

ωT 0=2 πk, where k=0 , ±1 , ±2 ,⋯ (1.34)

This shows that ω must be an integer multiple of ω0, i.e., the fundamental frequency. We can therefore, write

ϕk ( t )=e j kω0 t, where k=0 , ±1 , ±2 ,⋯ (1.36)

This is called the k-harmonic of the complex exponential signal.

General Complex Exponential Signals

The general complex exponential signals are of the form

x (t )=C eat

Where both C and a are complex numbers. Let us represent them as

C=|C|e jθ

a=r+ jω0

Thus C is written in polar form and a is written in Cartesian form, then

x (t )=C eat=|C|e jθ e(r+ jω0)t=|C|ert e j (ω0 t+θ) (1.42)





Using Euler’s formula, it can be written as

x (t )=C eat=|C|ert cos(ω0 t+θ)+ j|C|ert sin (ω0 t+θ) (1.43)

1 For r=0, the real and imaginary parts of a complex exponential are sinusoidal.2 For r>0, they correspond to sinusoidal signals multiplied with growing exponential.3 For r<0, they correspond to sinusoidal signals multiplied with decreasing exponentials.

This is shown graphically on the next page.





x(t)= |C| ertcos(0 t+)

t

x(t)= |C| e-rtcos(0 t+)

t





1.2.4 Periodic Signals (Recap)

A periodic continuous-time signal x (t) is defined as

x (t )=x (t+T )

where T is a positive number called the period.

A typical example is that of a sinusoidal signal x (t )=sin (t) for −∞<t<+∞. This is shown in the figure below:

For the above signal, the period is T=2π . It can be noticed that for any time t :

sin (t+2π )=sin(t)

Also

sin ( t+m2π )=sin( t)

where m is an integer.

Periodic signals are defined analogously in discrete-time.

The fundamental period T 0 of x (t) is the smallest positive value of T for which the equation x (t )=x (t+T ) holds.

A discrete-time signal x [n] is periodic with period N , where N is a positive integer, if it is unchanged by a time-shift of N , i.e., if

x [n ]=x [n+N ]

for all values of n.

The fundamental period N 0 of x [n] is the smallest positive value of N for which the equation x [n]=x [n+N 0] holds.





An example of a discrete-time periodic signal is given in the following diagram.

1.2.5 Even and Odd Signals (Recap)

A signal x (t) or x [n] is defined as an even signal if it is identical to its time-reversed counterpart, i.e., with its reflection about the origin.

Even continuous-time Signal x (−t )=x( t)Even Discrete-time Signal x [−n]=x [n]

A signal x (t) or x [n] is defined as an odd signal if,

Odd continuous-time Signal x (−t )=−x (t)Odd Discrete-time Signal x [−n ]=−x [n]

As a special case, the odd signal must be zero at t=0 or n=0. Examples of odd signals are given in the following figures.

x (t)

−6 π −4π −2π 0 2π 4 π 6 π t



A Discrete-Time Signal

Even Part Odd Part



An important fact is that any signal (continuous-time or discrete-time) can be broken into a sum of two signals, one of which is even and one of which is odd.

Signal Component Mathematical Form

Continuous-time Signalx (t)Even Part E v {x ( t ) }=12 [x ( t )+x (−t)]

Odd Part Od {x (t ) }=12 [ x (t )−x (−t)]

Discrete-time Signalx [n]Even Part E v {x [n]}=12 [x [n]+x [−n] ]

Odd Part Od {x [n]}=12 [x [n]−x [−n]]





1.3.2 Discrete-Time Complex Exponential and Sinusoidal Signals

A discrete-time complex exponential signal or sequence x [n] can be written as

x [n]=C α n (1.44)where C and α are, in general, complex numbers. This could also be written as

x [n]=C eβn (1.45)where

α=eβ

The form given in equation (1.44) is more often used in discrete-time signal processing.

Real Exponential SignalsIn this case both C and α are real numbers, and x [n] is called a real exponential. We see different types of behaviors, as shown below in the figure.

Discrete-time Real Exponential with >1x[n]=C n, C=1, =1.1

n

Discrete-time Real Exponential with 0<<1x[n]=C n, C=1, =0.9

n





Sinusoidal Signals

Another important complex exponential is obtained by using the form given in equation 1.45, and by considering β to be purely imaginary (so that |α|=1). More, specifically, we consider:

x [n]=e jω0n (1.46)

A closely associated signal is the sinusoidal signal

x [n]=A cos (ω0n+ϕ ) (1.47)

Using Euler’s formula, the complex exponential of equation (1.47) can be written as

x [n]=e jω0n=cos (ω0n )+ jsin (ω0n) (1.48)

Discrete-time Real Exponential with -1<<0x[n]=C n, C=1, =-0.9

n

Discrete-time Real Exponential with <-1x[n]=C n, C=1, =-1.1

n





Similarly, the sinusoidal signal of equation (1.25) can be written as

A cos (ω0n+ϕ)=A ( e j (ω0n+ϕ )+e− j (ω0n+ϕ)

2 )= A2e jϕ e jω0n+ A

2e− jϕe− jω0n

(1.49)

It can noted that the two exponentials in equation 1.27 have complex amplitudes. If we define Re {c } as the real component of a complex number c, and I m {c } as the imaginary part of c, then

A cos (ω0n+ϕ)=A Re {e j (ω0n+ϕ) }A sin(ω0n+ϕ )=A Im {e j (ω0n+ϕ )}

(a) x[n]=cos(2 n/12)

n

(b) x[n]=cos(8 n/31)

n

(c) x[n]=cos(n/6)

n





General Complex Exponential Signals

The general discrete-time complex exponential signals are of the form

x [n]=C α n

where both C and α are complex numbers. Let us represent them as

C=|C|e jθ

α=|α|e jω0

Thus C and α are written in polar form, then

x [n]=C α n=|C|e jθ|α|n e jω0n=|C||α|n e j (ω0n+θ)

Using Euler’s formula, it can be written as

x [n]=C α n=|C||α|ncos (ω0n+θ)+ j|C||α|n sin(ω0n+θ) (1.50)

4 For |α|=1, the real and imaginary parts of a complex exponential are sinusoidal.5 For |α|>1, they correspond to sinusoidal signals / sequences multiplied with growing

exponential.6 For |α|<1, they correspond to sinusoidal signals / sequences multiplied with decreasing

exponentials.

This is shown graphically on the next page.

(a) x[n]=cos(2 n/12)

n

(b) x[n]=cos(8 n/31)

n

(c) x[n]=cos(n/6)

n





Growing Sinusoidal Signal

n

Decaying Sinusoidal Signal

n





1.3.3 Periodicity Property of Discrete-Time Complex Exponential

There are many similarities between continuous-time and discrete-time signals. But also there are many important differences. One of them is related with the discrete-time exponential signal e jω0n.

The following properties were found with regard to the continuous-time exponential signal e jω0 t

:

1. The larger the magnitude of ω0, the higher is the rate of oscillations in the signal;2. e jω0 t is periodic for any value of ω0.

Let us now see how these properties are different in the discrete-time case:

1. To see the difference for the first property, consider the discrete-time complex exponential:

e j (ω¿¿0+2π )n=e j 2πn e j ω0n=e jω0 n¿ (1.51)

This shows that the exponential at ω0+2π is the same as that at frequency ω0. This is very different when compared with the continuous-time exponential case, in which the signals e jω0 t are all distinct for distinct values of ω0.

In discrete-time, these signals are not distinct. In fact, the signal with frequency ω0 is identical to signals with frequencies ω0±2π , ω0±4 π and so on. Therefore, in considering discrete-time complex exponentials, we need only consider a frequency interval of size 2π . The most commonly used 2π intervals are 0≤ω0≤2π or the interval −π ≤ω0≤ π.

Due to equation 1.51, as ω0 is gradually increased, the rate of oscillations in the discrete-time signal does not keep on increasing. If ω0 is increased from 0 to 2π , the rate of oscillations first increase and then decreases. This is shown in the figure 1.27 below.

Note in particular that for ω0=π or for any odd multiple of π,

Growing Sinusoidal Signal

n

Decaying Sinusoidal Signal

n





e jπn=(e jπ )n=(−1 )n (1.52)so that the signal oscillates rapidly, changing sign at each point in time (see figure 1.27).

2. The second property is the periodicity of the discrete-time complex exponential signals.

e jω0 (n+N )=e j ω0n (1.53)Or equivalently,

e jω0 N=1 (1.54)

0 10 20 300

0.5

1x[n] = cos(0*n)

0 10 20 30-1

0

1x[n] = cos( n/8)

0 10 20 30-1

0

1x[n] = cos( n/4)

0 10 20 30-1

0

1x[n] = cos( n/2)

0 10 20 30-1

0

1x[n] = cos( n)

0 10 20 30-1

0

1x[n] = cos(3 n/2)

0 10 20 30-1

0

1x[n] = cos(7 n/4)

0 10 20 30-1

0

1x[n] = cos(15 n/8)

0 10 20 300

0.5

1x[n] = cos(2 n)

Figure 1.27: Discrete-time sinusoidal sequences for several different frequencies





Equation 1.54 could only be true if ω0N is a multiple of 2π

ω0N=2πm (1.55)Or

ω0

2π=mN

(1.56)

This equation (1.56) means that the discrete-time signal e jω0n is periodic only when ω0

2π is a rational number (i.e. it could written as a fraction).

Table 1.1 Comparison of the signals e jω0 t and e jω0n.

e jω0 t e jω0n

Distinct signals for distinct values of ω0. Identical signals for values of ω0 separated by multiples of 2π .

Periodic for any choice of ω0. Periodic only if ω0=2πm /N for some integer N>0 and m.

Fundamental frequency ω0. Fundamental frequency ω0/m.Fundamental period

ω0=0: undefined

ω0≠0 :2πω0

Fundamental periodω0=0: undefined

ω0≠0 :m( 2πω0 )



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