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Signals and Systems:
Material for the classes on: 2/10/ 2/14/ 2/16/
The goals of the following three classes are:
Define and explore various types of signals Explore the concept of a system and define LTI systems Explore time and frequency domain representation of signals Review Fourier series/transform. Focus on their physical/practical significance Sampling and Nyquist rates. The phenomenon of aliasing. Numbering systems Conversion between types of signals
A signal represents a set of one or more variables and is used to convey the characteristic information (or the attributes) of a physical phenomenon.
The world around us is full of signals. Indeed our connection with the world is through the various signals that our senses can interpret for their corresponding physical phenomena: the human voice, the sounds of nature, the light we see, the heat we feel, are all signals. The classification of a signal is based on: (1) how is it represented in time and (2) how is its amplitude allowed to vary.
There are four basic types of signals based on the above classification. They are:
Continuous time, continuous value.
- Defined for each instant of time and its amplitude may vary continuously with time and assume any value. o Signals from transducers o Analog signals
Discrete time, continuous value.
- Defined at discrete instants of time and its amplitude may vary continuously with time and assume any value
Continuous time, discrete value.
- Defined for each instant of time and its amplitude may assume discrete values. o Signal is sampled at discrete times and the output assumes discrete values
Discrete time, discrete value.
- Defined at discrete instants of time and its output may assume discrete values o Digital signals
Before proceeding let’s define and calculate some of the most relevant parameters describing a signal.. The most frequently encountered signal, the generic sinusoidal signal, is given by the function,
x t ( ) = Α sin( ω t + φ) (1.2)
In the study of electronics we encounter this signal very frequently where x(t) may represent a voltage, a current or energy.
The parameters describing the signal of Eq. (1.2) are: A - the amplitude,^ ω^ - the radian
frequency, and φ - the phase.
The radian frequency ω is given in units of radians/sec and is related to the frequency
f given in cycles/sec. or Hz by
ω = 2 π f (1.3)
The period T of the signal is
1 2 T f
The phase φ represents a “shift” of the signal relative to origin ( t = 0 ).
Figure 2 illustrates the various parameters just described in a graphical fashion.
2
t (sec)
φ=
φ=π/
Figure 2. Sinusoidal signal with a phase of 0 degrees and 60 degrees.
In many applications involving time-varying signals, the relevant measurement parameters might be an average values of the signal. The electrical signal delivering the standard 120 Volt household electricity is a good example. The household electrical signal is a sinusoid with a frequency of either 60 or 50 Hz depending on location. The 120 Volts correspond to an average value of the signal and not to its amplitude. Figure 3 shows the typical 120 Volt signal measured at a wall outlet. Note that the amplitude of the signal is 170 Volts, not 120 Volts. So where does this number 120 Volts come form? It is certainly not a simple average since that would be zero for a signal symmetric about zero. 120 Volts is a number which gives an indication of the fluctuations of the signal about the average value. It is called the root-mean square value of the signal and as we
will see later when we study electrical signals in detail, it is important since it is related to energy content of the signal. The root-mean square value of a signal V t ( )is defines as
2 0
T Vrms = (^) T ∫ V t dt (1.5)
For V t ( ) = cos( ω t ), Vrms is calculated as follows.
2
2
2
2 2 0
2 2 0 2 0 2 2 0
cos ( )
cos ( ) 2
1 cos(2 ) 2 2
1 cos(2 ) 2 2 2
T rms
zero
V t dt T
t dt
t dt
t dt
π ω
π ω
π ω
∫
∫
For our electricity example, Vrms = 120 Voltsand thus the amplitude of the
corresponding sinusoidal signal is 170 Volts as indicated on Figure 3.
200
0
100
t (sec)
0 0.01 0.02 0.03 0.04 0.0 5
RMS
average
Figure 3. 120 Volt electrical signal
In some situations certain signals may prevent others from been received and understood. For example, our ability to listen to a conversation may be compromised by the engine noise of a low flying airplane or a by a passing train. In these situations the signals are still transmitted and received by our auditory system but we are unable to extract the
Systems.
Signals are always associated with one or more systems. For example, a certain system may generate the signal while another may operate on it in order to process it or to extract relevant information from it. The representation of a system with its associated input and output signals is shown on Figure 5. The input signal is also called the excitation signal and the output is also called the response signal. The system may thus be represented by an operator F which may be designed to perform any desirable operation on the input
signal x ( ) t resulting in the output signal y t ( ). In electronics, for example, the system
may be an amplifier where the excitation input voltage is operated on by the
operator F to produce the output with an amplification
vin ( ) t vout ( ) t A such that
vin ( ) t ⎯⎯→ vout ( ) t = Avin ( ) t
F
System x(t) y(t)
Input Output F^
x(t) y(t)
Figure 5. Block diagram of a system
Some common forms of the operator F are shown on the following table.
Integral ⎯⎯→ x t ( )^^ ⎯⎯ y t ( )⎯→
0
t
y t = ∫ x τ d τ
Amplifier ⎯⎯→ x t ( )^^ A ⎯⎯ y t ( )⎯→ y t ( )^^ = Ax t ( ) Multiplier (^) ⎯⎯→ ⊗ ⎯⎯ x t ( )^^ y t ( )⎯→ y t ( )^^ = x 1^ ( ) t x^2^ ( ) t Adder (^) ⎯⎯→ ⊕ ⎯⎯ x t ( )^^ y t ( )⎯→ y t ( )^^ =^ x 1^ ( ) t^^ + x 2^ ( ) t
The characteristics of the System operator F are fundamental in system analysis. We are
particularly interested in linear, time invariant (LTI) systems.
A linear system is one which is both homogeneous and additive.
A homogeneous system is one for which a scaled input voltage produces an equally scaled output voltage. Figure 6 illustrates the principle of homogeneity where can be any constant.
m
F
mx(t) my(t)
Figure 6. Homogeneous system
An additive system is one for which,
F
x (t) 1 y (t) 1
F
x (t) 2 y (t) 2
F
x 1 (t)+ x 2 (t ) y (t)+ 1 y 2 (t)
Figure 7. Demonstration of system additivity.
The general definition of a linear system is one that can be homogeneous and additive.
If y 1 ( ) t is the response of a system to an input x 1 ( t )and y 2 ( t ) is the response of a
system to an input x 2 ( ) t then if the system is linear the response to the signal
ax 1 (^) ( ) t + bx 2 ( ) t , where a and b are any constants is ay 1 (^) ( ) t + by 2 ( ) t. This very important
property of linear systems is called the principle of superposition which we may represent mathematically as
ax 1 (^) ( ) t + bx 2 (^) ( ) t ⎯⎯→ ay 1 (^) ( ) t + by 2 ( ) t (1.8)
F
In out study of electronic systems we will make extensive use of this property in order to obtain solutions of what seemingly appear difficult problems.
A time invariant system is one for which a delay τ 0 in the application of the excitation
signal (input) results in the same delay in the response signal (output).
For example if an input signal, x ( t ), to a system described by the operator F results in
the output y t ( )like,
x ( ) t ⎯⎯→ y ( )
F
t
Then the system is time-invariant if
x t ( − τ 0 ) ⎯⎯→ y t ( − τ
F
The interconnections between systems is also a very important consideration for their overall behavior. In electronic systems special attention is paid to their input and output characteristics. When systems are connected together the output characteristics of a system must “match” the input characteristics of the system that it connects to. As an
The microphone is a transducer which may be considered as a system that converts the pressure variations in the input signal X(t) to the voltage signal V1(t). In turn V1(t) is processed by the electronics module resulting in the signal V2(t). The electronics module may perform such operations as amplification, filtering and offsetting. Both signals V1(t) and V2(t) are time continuous analog signals. Signal V2(t) is in turn operated by module ADC resulting in signal V3(t) which is now a digital signal (discrete time) that may be further processed by the computer.
The conversion of the analog signal V2(t) to the digital signal V3(t) involves three very important operations: (1) sampling, (2) quantization and (3) encoding. Sampling is the process by which the signal values are acquired at discrete points in time. This is a non-linear process since information is irrevocably lost. Quantization is the process by which the continuum of amplitude values is converted to a finite number of values (quantized values). This is a non-linear process since information is irrevocably lost. Encoding is the process of converting each quantized value to a binary number represented by a binary bit pattern. No information is lost in this translation.
We will explore these operations in later sections. For now let’s establish the framework for signal representation and analysis.
Time and frequency domain
Physical signals, such as the voltage output of a microphone or the electrical signal output of a strain or a pressure gage, are usually represented as function of time. These signals may be manipulated (amplified, filtered, offset etc.) in the time domain and many applications deal with signals solely in the time domain.
However, it is often convenient and frequently necessary, when signal analysis and processing is required, to represent the signal in the frequency domain. A signal in the frequency domain shows “how much” of the signal is associated with a certain frequency. Figure 10 shows the time domain and the frequency domain representation of a sinusoidal signal with a frequency of 1kHz. Since this is a signal with a single frequency of 1 kHz, the frequency domain representation of the signal is a single line at a frequency of 1kHz. The height of the line at the frequency of 1 kHz corresponds to the magnitude or strength of the signal at that frequency.
As another example consider the signal given by the function
x( t ) = 1 + cos(1000 π t ) + 2 sin(600 π t ) (1.11)
This signal is plotted on Figure 11. The two frequencies present in the resulting signal are 500Hz and 300Hz.Therefore, in the frequency domain representation only these two frequencies contain signal information as shown on Figure 11. Note the strength of the signal as represented in the frequency domain.
2
0
1
t (sec)
0 0.001 0.002 0.003 0.00 4
2
0
1
Frequency (Hz)
0 500 1000 1500 2000
Figure 10. Time and frequency domain representation of a sinusoidal signal.
4
0
2
t (sec)
0 0.01 0.0 2
2
0
1
Frequency (Hz)
0 250 500 750 1000
Figure 11. Time and frequency domain representation of the signal
The graphical representation of signals in the frequency domain just presented will be enhanced by the appropriate mathematical representation of signals in the frequency domain. The theory of complex numbers is essential in understanding frequency domain representation. In the following section the concepts of Fourier analysis will provide us with a very powerful tool for the general transformation of a signal from the time domain to the frequency domain and equivalently from the frequency domain to the time domain.
Complex number arithmetic: A review A complex number may be represented in rectangular form as follows:
c = a + jb Rectangular format of complex number (1.12)
The number j = − 1. a is the real part of the complex number and b is the imaginary part
of the complex number.
The complex conjugate of a complex number is obtained by replacing with. For
the number given by Eq. (1.12) is c a
j − j ∗ (^) = − jb
The magnitude of the complex number is
magnitude = cc^ *^ = ( a + jb )( a − jb ) = r = a^2 + b^2 (1.13)
And the phase is
phase tan 1 b a
θ −^
The graphical representation of the complex number in the complex plane is:
Re
Im
a
b
θ
r
Euler’s identity is an important relationship in the theory of complex numbers. It states:
e j^ φ^ = cos φ + j sinφ (1.15)
From the graphical representation of a complex number and Euler’s identity we may represent the complex number in polar form as
c = r (cos θ + j sin θ) = rej^ θ Polar format of complex number (1.16)
Impulse Function. A review
In science and engineering there are many examples when an action occurs at an instant in time or at certain point in space. For example the force exerted on a baseball when it is hit by a bat is of very short duration. Also, the point test used in materials testing applies a very localized force on a material. The mathematical representation of this type of action is
2
2
2
t
t
t
ε
2
ε ε ε ε
For which we also impose the condition:
δ ε ( ) t dt 1
+∞ −∞ ∫ = (1.18)
The function may be thought of as a rectangular pulse of width ε and height 1/ε as shown
on Figure 13(a). In the limit ε → 0 , the height 1/ε increases in such a way that the total
area is 1. This leads to the definition
0 ( ) t lim (^) ε( ) ε δ δ t →
The function δ ( ) t is called the unit impulse function which is also known as the Dirac
Delta function or simply as the Delta function. The graphical representation of the Delta function is shown on Figure 13(b)
δ(t)
1/ε
(a)
t
δ(t)
1
(b)
Figure 13. Delta function (a) visualization and (b) symbol
For a more general representation, the function δ ( t − t 0 )represents is shifted Delta
function and represents an impulse centered at t = t 0. The graphical and mathematical
representations of this general Delta function is,
t
δ(t)
t 0
0
0 0
( ) 0 for
t dt
t t
δ τ
δ τ τ
+∞ −∞
∫ (^) (1.20)
The usefulness of the Delta function results not from what it represents but rather from what it can do. The two fundamental properties, and default definitions, of the Delta function are:
(^220) ( 0 ) δ τ τ ∞^ e j^ π^ f τ e − j^^ π^ f τ df − ≡ (^) ∫−∞ (1.21)
f t ( ) δ ( t τ 0 ) dt f ( 0 )
∞ −∞ ∫ −^ =^ τ^ (1.22)
Equation (1.22) is referred to as the sampling property of the Delta function and it is a very important property used extensively in signal analysis.
(a) (b)
Figure 14. (a) the time domain signals of a middle C note of a piano represented as a voltage from a microphone. (b) Fourier transform of the signal represents the same signal in the frequency domain.
Before proceeding with the physical and thus the practical significance of FT let’s become more familiar with the process by calculating the transform for various practical signals. We will look at periodic as well as non-periodic signals. Let’s start with the calculation of the Fourier transform of the signal
v t ( ) = sin( ω 0 t ) (1.27)
This is our familiar sine wave characterized by a frequency of 2 πω 0. Since this signal
represents - by definition - a single frequency, we anticipate that in the frequency domain, all information will be contained at that frequency. So let’s proceed with the calculation to determine the Fourier transform of v t ( )which is given by
V ( ω) sin( ω 0 t ) j^^ ω t (1.28)
∞ (^) − = (^) ∫−∞ e dt
By using Euler’s identity, Eq. (1.15), we obtain,
( )
0 0
( 0 ) ( 0 )
j t j t j t
j t j t
e e V e j
j e e d
ω ω ω
ω ω ω ω
∞ (^) − − −∞ ∞ (^) − + − − −∞
∫
dt
t
dt
According to Eq. (1.21),
2 π δ ω( ω 0 ) e j^ (^ ω^ ω^0 ) t (1.30)
∞ (^) − + −∞
And,
2 π δ ω( ω 0 ) e j (^^ ω^ ω^0 ) t (1.31)
∞ (^) − − −∞ − = (^) ∫ dt
Therefore, Eq. (1.29) becomes
V ( ω ) = π j [ δ ω( + ω 0 ) − δ ω( − ω 0 )] (1.32)
The graphical representation of V ( ω) is shown on Figure 15.
V j( ω)
ω 0 ω
π/j
π/j
Figure 15. Fourier transform of a sine wave.
Similarly the Fourier transform of the signal
v t ( ) = cos( ω 0 t ) (1.33)
Is calculated as follows
( )
[ ]
0 0
0 0
0
( ) ( )
0 0
( ) cos( )
j t
j t j t j t
j t j t
V t e dt
e e e dt
e e
ω
ω ω ω
ω ω ω ω
∞ (^) − −∞ ∞ (^) − − −∞ ∞ (^) − − − + −∞
∫
∫ dt
Figure 16 shows the Fourier transform of the cosine signal.
V (ω)
π π
Figure 16. Fourier transform of a cosine wave.
