Contents
Introduction to Signals and Systems.
CONTINUOUS-TIME SIGNALS.
SYSTEMS.
Basic continuous-time signals.
Unit-Step Function, Unit-Impulse (Dirac Delta) Function, Ramp Function.
Unit-Step Function.
Unit Impulse Function.
Relationship between u(t) and d (t):
Unit-Impulse Properties.
Ramp Function.
Relationship between r(t) and u(t):
Exponential Signals:
Sinusoidal Signals:
Periodic signals:
Modifications and transformations of the independent variable t and Signal symmetries
Modifications and transformations of the independent variable t.
Signal Symmetries
Continuous-Time Convolution
Convolution Properties. 31
—————————————————————————————
1.1 Introduction to Signals and Systems
CONTINUOUS-TIME SIGNALS
A signal is a function of one or more independent variables that contains information about the behaviour or nature of some phenomenon
Figure 1.1: A signal with main parameters
Continuous-time signals are functions of a real argument x(t) where t can take any real value.
Figure 1.2: Example of Continuous-time signal
Figure 1.3: Example of Continuous-time signal
A discrete-time signal is a function of an argument that takes values from a discrete set x[n] where n Î {…-3, -2,-1,0,1,2,3…}
Figure 1.4: Example of Discrete -time signal
Discrete-time signal can be obtained by taking samples of an analogue signal at discrete instants of time. The values for x may be real or complex
Square brackets are used to denote a discrete-time signal x[n] to distinguish between the continuous-time and the discrete-time signals
n Electrical signals
– Voltages and currents in a circuit
Figure 1.5: Voltage and current in a circuit
n Acoustic signals
– Acoustic pressure (sound) over time
Figure 1.6: Acoustic pressure over time signal
n Mechanical signals
– Velocity of a car over time
Figure 1.7: Velocity of a car over time signal
n Video signals
– Intensity level of a pixel (camera, video) over time
Figure 1.8: Video signal
Example: Signals in Electrical Circuit
The signals v_{out} and v_{in} are patterns of variation over time
SYSTEMS
q A system is a transformation from one signal (called the input) to another signal (called the output or the response).
q The figure Shows the block diagrams of continuous-time and discrete-time systems with input signal x and output signal y
Figure 1.9: Continuous and Discrete-time signals.
Types and examples of systems
Continuous-time systems:
y(t) = x(t) + x(t-1) and y(t) = x^{2}(t)
– Discrete-time systems:
y[n] = x[n] + x[n-1] and y[n] = x^{2}[n]
q Multiple input and/or output signals are also possible. In this case the system is referred to as a multiple-input, multiple-output (MIMO) system as shown in Figure.
Figure 1.10: MIMO system
Practical Examples of systems
– A circuit involving a capacitor can be viewed as a system that transforms the source voltage (signal) to the voltage (signal) across the capacitor
– A CD player takes the signal on the CD and transforms it into a signal sent to the loud speaker
A communication system is generally composed of three sub-systems, the transmitter, the channel and the receiver
Example: Communication System:
Communication system contains three subsystems: Transmitter, Channel and Receiver.
Figure 1.11: Communication system
1.2 Basic continuous-time signals
In this chapter, we will define the following basic continuous-time signals to be utilized in our studies:
- Unit-Step Function.
- Unit-Impulse (Dirac Delta) Function.
- Ramp Function.
- Exponential Signals.
- Sinusoidal Signals.
- Complex exponential and damped sinusoid and explained the main properties of above-mentioned signals.
Part 1: Unit-Step Function, Unit-Impulse (Dirac Delta) Function, Ramp Function
Unit-Step Function
n The continuous-time unit step function u(t) is defined by:
(1.1)
n Note that u(t) is discontinuous at t=0.
Figure 1.12: Unit-step function
n Example : the rectangular pulse shown in Fig.2 can be expressed as:
Figure 1.13: Rectangular pulse
Unit Impulse Function
The unit-impulse (Dirac delta) function is defined as follows:
(1.2)
The “1” written beside the arrow indicates that the “area” of the impulse is unity.
Figure 1.14: Unit-impulse function
Relationship between u(t) and d (t):
δ(t) is the first derivative of u(t):
(1.3)
And u(t) can be expressed as the running integral:
(1.4)
Example: A given signal x(t) and its derivative
The derivative of x(t) is zero except at the discontinuities
Unit-Impulse Properties
The scaled-impulse function kδ(t) is the derivative of the scaled unit step ku(t):
(1.5)
Figure 1.15: The unit-impulse function
Example:
Sampling property:
(1.6)
Figure 1.16: the sampling property example
Sifting property:
(1.7)
In general,
(1.8)
Ramp Function
The continuous-time unit ramp function shown in figure is defined by:
(1.9)
Figure 1.17: The ramp function
Equivalently, we may write:
(1.10)
Relationship between r(t) and u(t):
The unit step function is the first derivative of the ramp function:
(1.11)
And the ramp function is obtained by integrating the unit step function:
(1.13)
Example: The signal x(t) and its derivative.
Exponential Signals:
The exponential signal is given by:
(1.14)
where A and λ are generally complex numbers.
Special cases of exponential signals:
- If A and λ are real: (Real exponential)
- If λ is purely imaginary: (Complex exponential or sinusoidal)
- If both A and λ are complex: (Complex exponential or damped sinusoid)
Real exponential: (A and λ are real)
Three cases can be distinguished in the real exponential
Case 1: If λ> 0, x(t) increases exponentially with t
Example:
Case 2: If λ < 0, x(t) decreases exponentially with t
Example:
Case 3: If λ= 0, x(t) is a constant (dc) signal
Example:
Sinusoidal Signals:
Complex exponential or sinusoidal: If λ is purely imaginary , then
Using Euler’s formula |
(1.15)
(1.16)
(1.17)
(1.18)
Figure 1.18: The real sinusoid
Periodic signals:
- Any continuous-time signal x(t) that satisfies the condition
x(t+T) = x(t) , for all t
where the smallest positive value of T known as the fundamental period of the signal x(t), is classified as a periodic signal.
- A signal x(t) that is not periodic is referred to as an aperiodic signal.
- In the case of sinusoidal signal, x(t) is a periodic signal, that is
x(t+T) = x(t) for all t
where the smallest positive value of T is called the period of the signal x(t)
- The period T is given by:
(1.19)
where f0 is the frequency of the signal in Hertz.
Examples:
Complex exponential or damped sinusoid: If A and λ are complex,
Then
Figure 1.19: Growing sinusoid (r >0) (b) Damped sinusoid (r <0)
Modifications and transformations of the independent variable t and Signal symmetries
Modifications and transformations of the independent variable t
Many simple and important signal processing operations can be described mathematically by modifications or transformations of the independent variable t.
- Time Reversal
- Time Scaling
- Time Shifting
- Combinations
Time Reversal
The modified signal
is the time-reversed version of x(t), as depicted in figure1.20
Figure 1.20: Signal x(t) and its time reversal x(-t)
Note that y(1)=x(-1), y(-6)=x(6), etc., for all values of t.
Time Scaling:
The signal
(1.20)
is a time scaled version of the signal x(t),
Where a is a real constant.
- if |a|>1 then y(t) is a time-compressed or speeded-up version of x(t), as depicted in Fig.2
- If |a|<1 then y(t) is a time-stretched or slowed-down version of x(t), as depicted in Fig.2
Figure 1.21: Time Scaling
An important special case is the time-scaled impulse function δ (t/a), which is satisfy
Example:
Time Shifting
The signal
is a time–shifted version of the signal x(t).
Where t0 is a constant.
- if t0 > 0, x(t) is shifted to the right by t0 units (delayed in time), as depicted in Fig.1.22
- if t0 < 0, x(t) is shifted to the left by t0 units (advanced in time), as depicted in Fig.1.22
Figure 1.22: Time Shifting
Combinations
The three transformations in time are of the general form
(1.21)
where a and b are real constants.
It is important to develop the facility to perform these transformations correctly.
A general approach for transformation of the variable t is as follows
We can perform the transformation of the form x(at+b) as follows:
Step(1): Plot x(at)
Step(2): Shift x(at) by b/a units to get x(a(t+b/a)).
Example:
Given the signal x(t) as shown in figure 4
a- Sketch the transformed signal y(t)=x(-2t+6).
b- Sketch the transformed signal w(t)=x(2t+4).
Figure 1.23: Time reversed and time scaled signal
Solution:
a- To sketch y(t) = x(-2t+6) = x(-2(t-3))
Step(1): Plot x(-2t)
Step(2): Shift x(-2t) to the right by 2 units to get x(-2(t-3))
b- To sketch w(t) = x(2t+4)= x(2(t+2))
Step(1): Plot x(2t)
Step(2): Shift x(2t) to the left by 2 units to get x(2(t+2))
Signal Symmetries:
A real-valued signal x(t) is said to be:
Even if x(t) = x(-t)
Odd if x(t) = – x(-t) and x(0) = 0
An even signal is one which is symmetric about the y-axis (vertical axis through the origin)
An odd signal is one which is symmetric about the origin
Examples: (Even Signals)
Examples: (Odd Signals)
In general, any real-valued signal x(t) can be written as the sum of its even and odd parts:
(1.22)
We conclude that the even part for any odd signal is zero and the odd part of any even signal is zero.
Example: Find the even and odd parts for each of the following signals:
Solution:
The even and odd parts of the signal x(t)=u(t) are shown in Fig.1.24
Figure 1.24: The even and odd parts of the signal x(t)=u(t)
Continuous-Time Convolution:
Introduction:
Convolution is a fundamental operation in signal-processing and system theory.
The convolution of two continuous-time signals x(t) and h(t) to produce a new signal y(t) is denoted by:
(1.23)
The symbol * is used to denote the convolution operation
A convolution is an integral that expresses the amount of overlap of one function h(t) as it is shifted over another function x(t).
The convolution of two continuous-time signals x(t) and h(t) to produce a new signal y(t) is defined by:
(1.24)
This equation referred to as the convolution integral
Figure 1.25: Convolution graphical examples
The animations above graphically illustrate the convolution of two boxcar functions (left) and two Gaussians (right). In the plots, the green curve shows the convolution of the blue and red curves as a function of t, the position indicated by the vertical green line. The gray region indicates the product
(1.25)
as a function of t, so its area as a function of t is precisely the convolution.
Continuous-time convolution satisfies the following Properties:
- Commutative property:
- Associative property
- Distributive property
Another interesting property is obtained by considering convolution with impulse function:
That is the convolution of an arbitrary signal x(t) with a shifted impulse d(t-t0) is the shifted signal x(t-t0).
In particular, for t0=0, we have simply:
For example
In this section, we will define and practice convolution purely as a mathematical operation so that we will be more comfortable with its application to linear systems
To compute the convolution y(t)= x(t)*h(t), it is often useful to graph the functions in the integrand of the convolution integral. This can help to determine the integrand and integration limits of the convolution integral.
The steps of this graphical aid to computing the convolution integral
are listed below:
1-Obtain and graph the signal h(t-t) from h(t) by a reflection about the origin and a shift by t.
2- Find the product x(t) and h(t-t) for all values of the dummy variable t, with t fixed at some value.
3- Integrate the product x(t) h(t-t) over all t to produce a single value y(t).
4- Steps 1,2, and 3 are repeated as many times as necessary until x(t)*h(t) is computed for all values of t from -∞ to ∞ to produce the entire function y(t).
Figure 1.26: Steps of performing convolution
Example: Find y(t)= x(t)*h(t) where x(t)=u(t) and h(t)= u(t)
Solution
- Graph x(t) and h(t)
- Replace t with t in x(t) and h(t)
- Graph h(- t)
- Graph h(t-t) for t < 0 :
For t < 0: the nonzero portions of x(t) and h(t-t) do not overlap, and thus
y(t) = 0
- Graph h(t-t) for t ≥ 0 :
For t ≥ 0: the nonzero portions of x(t) and h(t-t) do overlap from the beginning of x(t) at t = 0 to the end of h(t-t) at t = t, and thus
- Therefore the convolution of x(t)=u(t) and h(t)=u(t) produces the ramp function:
- Result: u(t)*u(t) = r(t)
Example 2: Find and sketch y(t)= x(t)*h(t)
Where x(t) = 2[u(t-2) – u(t-4)]
and h(t) = [u(t) – u(t-1)] – [u(t-1) – u(t-2)]
Solution:
- Graph x(t) and h(t)
- Replace t with t in x(t) and h(t)
- Convolution can be divided into 6 parts
I. For t < 2:
– Two functions do not overlap
– Area under the product of the functions is zero, then y(t)=0
II. For 2 £ t < 3:
– Part of x(t) overlaps part of h(t-t)
– Area under the product of the functions is
III. For 3 ≤ t < 4:
– Part of x(t) overlaps part of h(t-t)
– Area under the product of the functions is
IV. For 4 ≤ t < 5:
– Part of x(t) overlaps part of h(t-t)
– Area under the product of the functions is
V. For 5 ≤ t < 6:
– Part of x(t) overlaps part of h(t-t)
– Area under the product of the functions is
VI. For t ≥ 6:
– Two functions do not overlap
– Area under the product of the functions is zero, then y(t)=0.
- Result of convolution:
In the above examples we used the convolution integral
According to the commutative property, the convolution operation gives the same results if x(t) and h(t) are swapped:
Convolution Properties
There are many important properties of convolution that will be quite useful to us in our study of linear systems.
Assuming that the convolution integral
converges.
1. If x(t) is even and h(t) is odd, then y(t) is odd.
2. If x(t) and h(t) are both odd, then y(t) is even.
3. If x(t) and h(t) are both even, then y(t) is even.
4. If x(t) is periodic, then y(t) is periodic.
5. Letting Ax be the area under the curve x(t), that is and similarly for the areas Ah and Ay, then Ay =Ax Ah
6. Time scaling:
Examples:
7. Time reversal:
Examples
8. Time shifting:
Examples:
9. The convolution of two arbitrary right-sided signals x(t) and h(t), which are zero for t < t1 and t < t2, respectively, produces a right-sided signal y(t) that is zero for t < t1+t2 as illustrated in Fig.1.27
Figure 1.27: Convolution of two right-sided signals
10. The convolution of two arbitrary left-sided signals x(t) and h(t), which are zero for t > t3 and t > t4, respectively, produces a left-sided signal y(t) that is zero for t > t3+t4 as illustrated in Fig.1.28
Figure 1.28: Convolution of two left-sided signals
11. The convolution of two arbitrary finite-duration signals x(t) and h(t), which are nonzero for t1 < t < t3 and t2 < t < t4 , respectively, produces a finite-duration signal y(t) that is nonzero for t1 + t2 < t < t3 + t4 as illustrated in Fig.1.29
Figure 1.29: Convolution of two finite-duration signals