System Stability and Discretization

A transfer function is a mathematical representation of a system's input-output relationship.

With the help of a transfer function we can determine if a system is stable or unstable by looking at its poles.

> if all poles of a system are present on the left half plane the system will be stable —> the Output will be finit in stady state (time at which the system gives the final output)

>if only one pole is in the right side of the plane the system will be unstable —> Output at stady state is infinit.

> if the system has complex conjugated poles on the imaginary axes the system will be marginally stable —> output will oszillate between two finite values (e.g. sine wave)

Discretization is the process of converting a continuous-time signal or system into a discrete-time signal or system. It involves sampling the continuous-time signal at regular intervals and quantizing its amplitude values to a finite number of levels. The result is a sequence of discrete-time values that can be stored and processed using digital systems.

There are several reasons why discretization is necessary. First, many real-world signals and systems are inherently discrete in nature, such as digital audio or video signals. Second, digital systems are often more efficient and cost-effective than analog systems, making them a popular choice for signal processing applications. Finally, discretization allows for the use of digital signal processing techniques, which can provide greater accuracy and flexibility than analog techniques.

There are two main paths of discretization:

1. Analog-to-Digital Conversion (ADC): This involves converting a continuous-time signal into a discrete-time signal. The process involves sampling the signal at regular intervals and quantizing the amplitude values to a finite number of levels. The resulting sequence of discrete-time values can be stored and processed using digital systems.

2. Digital-to-Analog Conversion (DAC): This involves converting a discrete-time signal into a continuous-time signal. The process involves converting the digital values into analog voltage levels using a DAC. The resulting analog signal can then be processed by analog systems.

Overall, discretization is a crucial process in digital signal processing, allowing us to convert real-world signals into a format that can be easily processed by digital systems.

Problems:

> Loss of information —> values only at discrete rate

>Quatisation Error —> can be decreased by increasing the sampling frequency but that means more computing power is needed

>Quatising is adding delay since we have to sample data and quantize it using algorithm befor we can use the data.

Benefits:

>using digital signal processing such as microcontrolers or computers

>information loss can be handeld

>

Word!

The forward Euler method is a numerical algorithm used to approximate solutions to ordinary differential equations (ODEs). The method uses a first-order Taylor series expansion to approximate the solution at each time step.

To use the forward Euler method, we first need to know the initial condition of the ODE, which is the value of the solution at the starting time. Then, we break up the time interval we are interested in into a series of discrete time steps, with a fixed time step size, Δt.

At each time step, the algorithm uses the current value of the solution and the ODE itself to calculate an estimate of the solution at the next time step. The estimate is obtained by taking the current value of the solution and adding to it the product of the time step size and the derivative of the solution at the current time. Mathematically, the formula for the forward Euler method is:

y_{n+1} = y_n + f(t_n, y_n) * Δt

where y_n is the approximate solution at time t_n, f(t_n, y_n) is the derivative of the solution at time t_n, and y_{n+1} is the approximation of the solution at the next time step t_{n+1} = t_n + Δt.

It's worth noting that the forward Euler method is a first-order method, which means that the error of the approximation is proportional to the size of the time step squared. Therefore, a smaller time step size results in a more accurate approximation.

Overall, the forward Euler method is a simple and widely used numerical method for solving ODEs, but it may not always provide accurate results for certain types of ODEs or when the time step size is too large.

Backward Euler is a numerical method for solving ordinary differential equations (ODEs). It is a type of implicit method, which means that it uses information from future time steps to solve for the solution at the current time step.

The basic idea of the Backward Euler method is to approximate the derivative at a particular time point using the value of the solution at the next time point. In other words, we approximate the derivative at time t+Δt, rather than at time t. This leads to an equation that can be solved iteratively to find the solution at each time step.

The formula for the Backward Euler method is:

y(t+Δt) = y(t) + Δt*f(y(t+Δt))

where y(t) is the solution at time t, Δt is the time step, and f(y(t+Δt)) is the derivative of the solution at time t+Δt.

To use this formula, we need to have an initial condition for the solution at some starting time t_0. Then, we can iteratively apply the formula to find the solution at each subsequent time step.

One advantage of the Backward Euler method is that it is unconditionally stable, which means that it will always converge to the correct solution for a given ODE, regardless of the size of the time step. However, it can be computationally expensive, especially for stiff ODEs, because it requires solving a system of equations at each time step.