# Solution 2.39a: Integrators vs Differentiators

• stanigator
In summary, integrators are preferred over differentiators when representing a differential equation system in a block diagram due to the susceptibility of differentiator electronics to environmental noise. Additionally, differentiators can lead to unstable systems with certain bounded inputs, such as a step input resulting in an unbounded output with characteristics of an impulse.
stanigator
For the question (problem 2.39a) stated in the picture attached to this message, I don't understand why integrators should be used in favour of differentiator for the block diagram representing the differential equation system. Is this b/c of the nature of the differentiator electronics too susceptible to environmental noise?

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In addition to what you mention of the susceptibility to noise, another point comes to mind. The differentiator would represent an unstable system with regards to certain types of bounded inputs. An example of this would be a step input. This input would result in the unbounded output with the characteristics of an impulse. That is, the derivative of a unit step, $u(t)$, is in fact the unit impulse function, $\delta(t)$:

$$\frac{du}{dt} = \delta(t)$$

The choice between integrators and differentiators depends on the specific application and system being modeled. In general, integrators are used when the output of the system needs to be a cumulative sum of the input over time. This is useful for modeling systems that involve accumulation, such as in finance or population growth.

On the other hand, differentiators are used when the output of the system needs to be the rate of change of the input over time. This is useful for modeling systems that involve rates of change, such as in physics or engineering.

In the case of the block diagram representing a differential equation system, the use of integrators may be more appropriate because the output of the system is often a cumulative sum of the input over time. Additionally, the use of differentiators may introduce noise and instability in the system due to the amplification of high-frequency components in the input signal.

However, it is important to note that the choice between integrators and differentiators also depends on the accuracy and resolution required for the specific application. In some cases, a combination of both integrators and differentiators may be necessary to accurately model a system. Ultimately, the decision should be based on the specific requirements and characteristics of the system being modeled.

## 1. What is the difference between integrators and differentiators?

Integrators and differentiators are two types of mathematical operations commonly used in signal processing and control systems. Integrators perform the mathematical operation of integration, which is the reverse of differentiation. Differentiators, on the other hand, perform the mathematical operation of differentiation, which is the process of finding the rate of change of a signal over time.

## 2. When should I use an integrator vs a differentiator?

The choice between using an integrator or a differentiator depends on the specific application and the desired outcome. Integrators are commonly used to smooth out signals and eliminate noise, while differentiators are used to amplify high-frequency components of a signal.

## 3. How do integrators and differentiators affect the shape of a signal?

Integrators and differentiators both have a significant impact on the shape of a signal. Integrators tend to "flatten" out a signal, making it smoother and less jagged, while differentiators can make a signal more "peaky" or "spiky" by amplifying high-frequency components.

## 4. Can I combine integrators and differentiators in a single system?

Yes, it is possible to combine integrators and differentiators in a single system to achieve a desired outcome. In fact, many control systems use a combination of both operations to achieve the desired response from a system.

## 5. Are there any drawbacks to using integrators and differentiators?

While integrators and differentiators are useful mathematical operations, they can also introduce instability in a system if not used carefully. They can also amplify noise and distortions in a signal if not properly designed and implemented. Therefore, it is important to carefully consider the application and design before using integrators and differentiators in a system.

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