Hello readers, Welcome to another tutorial about the signal and system. In this lecture, you are going to read details about the ramp response of a signal. In the past lectures, we have been dealing with different types of responses of LTI systems, and therefore, we know that linear invariant systems, or LTI systems, are those which follow the rules of linearity and are also time-invariant. So, at present, our focus is to examine what happens when the ramp signals are fed into the LTI system and which type of output signal we receive. Here is a glimpse at today’s topic that we will learn deeply.
What is the RAMP signal?
How can you define the ramp response?
How to use the ramp function in MATLAB to get the ramp response?
What are some important properties of the ramp function?
How is ramp response used in different fields in different ways?
What is a Ramp Signal?
We all know that a signal is a function of one or more variables that are independent and contain some information in them. When talking about ramp signals, we get the following definition:
“A ramp signal is the one that always has its initial condition at 𝑡 = 0, and with time, this signal increases exponentially. Therefore, it is linear in its behavior with time.“
So, when representing these with the help of graphs, we get a smooth result all the time instead of any abrupt change in the pictorial representation.
As with other types of signals, ramp signals can also be described in two ways:
Continuous-time signal
Discrete-time signals
When the ramp signal is in the form of continuous values, we represent them as:
r(t)= { t for t=>0 } and {0 for t<0}
Similarly, the same condition in the discrete-time format is described as
r(n)= { n for n=>0 } and {0 for n<0}
Let us clarify the meaning of the statements given above. In both cases, the value of t or n increases with time simultaneously. Therefore, we get the smooth slop in the case of continuous signals and smooth points when we are dealing with discrete ramp signals. In other words, the values on the x-axis and the y-axis for a ramp signal are always equal if we are plotting a graph for them.
What is the Ramp Response?
As we have discussed at the beginning of this lecture, linear time-invariant systems have the linearity property, and you can use the ramp function in the time-invariant system. It has the input at one end, then the input faces some procedures according to the conditions, and from the other side of this LTI system, we get the output. So, in simple words, we define the ramp response as:
“The ramp response is one of the responses of the LTI system when the signal used as input is the ramp signal and the output of that system has the same features.”
In some places, the ramp function is defined in other ways, but the basic definition remains the same as we have provided you before. Other ways to introduce the ramp function are:
The resultant value is when the mean is calculated between the independent variable and its absolute value, which is called the ramp response.
R(x)=(x+|x|)/2
Here,
R= ramp function
x = the variable based on which R is taken.
Ramp Response in MATLAB
Code:
num=[0.9 0.18 0.27];
den=[1 0.2 0.3 0.4];
n=0:0.1:7;
x=n.*(n>=0);
y=filter(num,den,x);
stem(n,y)
xlabel('n /The Engineering Projects.com')
ylabel('amplitude')
grid on
title('Ramp Response in CT')
Output:
We have used the filter function in the previous tutorials as well. But, for a revision, it is important to know about it again.
Filter Function in MATLAB
The filter function is used when the digital filters are to be applied to a vector in MATLAB. The information that is to be fed into this function is delayed before this function. The syntax is given as
filter(b,a,x)
Where
b=co-efficient of the numerator
a=co-efficient of the denominator,
And x is different in different conditions.
Filter returns the filtered data as a vector with the same size as x if x is a vector.
The filter operates along the first dimension of a matrix if x is one, and it returns the filtered data for each column.
If x is a multi-dimensional array, the filter operates along the first dimension of the array whose size is not 1.
In our case, num and den are the matrices. So we are using the second case.
Explaining Code for Ramp Response
Have you seen the code? For a super easy-to-understand, we are going to discuss every step in detail.
In the first step, we declared two arrays and stored their values in the num and den separately.
In the second step, the time is provided to MATLAB, which shows the time has an upper limit of seven and a lower limit of zero. The interval between the times is taken as 0.1.
Now, we have just used these two arrays for the multiplication and used the dot along with the multiplication sign so the compiler may understand that it has to multiply every term in the loop.
In the next step, we are going to use all the results in the filter function, and this is the step that provides us with the results in a better manner.
The results are then fed into the variable y.
In the end, to get the result graphically, we used the stem function and provided the values of n as time.
Note:
If you want to have the same graph in the continuous-time signal, you just have to:
Replace the n with t for the best representation.
Replace the stem function with the plot function.
In this way, you will get the same graph, but the slope will be smooth.
Properties of the Ramp Function
To deal with different types of cases when using ramp response, one must know the properties of ramp response in detail to avoid the long calculation all the time. So here are some of these:
Laplace Transform of Ramp Response
The Laplace transform is used to convert the system from the time domain to the frequency domain, and when talking about the ramp function, it says that the Laplace transform of a ramp function results in a calculation that is equal to the square of the variable based on which the integration of that ramp response occurs.
Observe that this property is valid for the single-sided Laplace transform.
Fourier Transform of the Ramp Function
The Fourier transform is another way to convert the function from the time domain to the frequency domain so that it may become easy to deal with that particular function. The process of Fourier transform on the ramp function provides us with the following results:
Non-Negative Nature of Ramp Function
At the start, we mentioned that the ramp function does not exist in negative values and that the values are always in positive coordinates. So, we can represent this property mathematically as
∀ x ∈ R: R(x)>=0
In this way, the definition and nature of the ramp response are now clearer in our minds. There are certain conditions when the negative values of the ramp function and in return, ramp response are obtained. In such conditions, the negative values are always ignored.
Derivative of Ramp Function
To know about this property, you must know about the Heaviside step function. It is the type of step function that always has a zero for negative values and one for positive values. It is usually denoted with a capital H. When we take the derivative of the ramp function, we get the Heaviside step function. When writing it mathematically, we get the following equation:
R’(x)= H(x) for x !=0
There are different conditions and properties of the Heaviside step function, but we are not going to explain them because it is out of the scope of this lecture. For now, you just have to remember the equation that we discussed just before this paragraph.
Applications of Ramp Response
We know the signals are important in almost every type of field of science and therefore, we have the emphasis learning more and more about them. When the topic is ramp response, there are different ways in which ramp signals are used, and therefore, ramp response also has an application in that particular field. So let’s discuss the important fields where ramp function and, in return, ramp response are used.
System Testing using Ramp Response
We know that in the ramp function, the values on the x-axis are equal to the y-axis and these are always positive values. Therefore, we can guess the results before the time. This property is easily used in different systems for testing different conditions or when starting a new system, it is used to check particular features.
Ramp Response in Engineering Fields
In different engineering fields, signals are used in different ways, as we are learning in one such subject, “Signal and System”. Electrical, computer, civil, and other engineering disciplines involve the use of ramp signals and ramp responses.
Ramp Response in the Filed of Finance
Those who belong to the field of finance know that the payoff of the call option is in the form of a ramp function, and therefore, ramp response has major usage in this field. The put option is set by flipping the ramp horizontally, and the short option is obtained when the ramp is flipped vertically. The graph formed in this way is called the "hockey stick" because of its shape.
It becomes easy to deal with the results, and we get accuracy.
Using Ramp Response in Statistics
When we use the ramp function, we face different conditions while studying statistics. Some cases are
Multivariate Adaptive Regression Splines or MARS
Hinge Function
To deal with such complex calculations, it becomes easy to use the ramp properties so that unnecessary work may be ignored. For example, the non-negative property of the ramp function provides us the facility to ignore the negative values and focus on the calculations of positive values.
Traffic Signaling and Ramp Response
It may look unusual, but the real-time application of the ramp function is in the traffic signals. A bottleneck is a situation when on the highway several vehicles are entered in an unmarred way and it results in the blockage of traffic. A ramp signal is used to break up this blockage, which is a cost-effective way to deal with this situation. The work is done. When the ramp signal is applied to the traffic lights effectively, this is one of the easiest ways to deal with this situation.
Today, we learned a lot about the ramp response, ramp function, and unit ramp signal. We saw the introduction of all of these and the interesting thing about the ramp response was the code of the ramp response in MATLAB because it was so clean and easy to understand. In addition to this, we have learned about some properties of the ramp function that helped us to clear the concepts and to know how we can skip long calculations all the time if we know the properties of the ramp function. In the end, we had a glance at the application of the ramp function in different fields of science, such as statistics, engineering, and finance. We hope it was an informative lecture for you. Stay with us for more interesting lectures about signals and systems.
