Question

Give a state space form description of a discrete-time system where \(U=Y=\\{0,1\\}\), such that the output \(y\) at time \(t\) equals 1 if the input until (and not including) \(t\) has shown an even number of 1 's and equals 0 otherwise.

Short answer

The state's space form has \(x(t+1) = x(t) \oplus u(t)\) and \(y(t) = 1 - x(t)\).

Step-by-step solution

Define the State

We need to establish a state that keeps track of the number of 1's in the input. Let's define a binary state variable \( x(t) \) that represents the parity (even or odd) of the number of 1's received so far. This state will be 0 if the number of 1's is even and 1 if it is odd.

State Transition Function

The state at time \( t+1 \) depends on the current state \( x(t) \) and the current input \( u(t) \). If \( u(t) = 0 \), the state remains the same, i.e., \( x(t+1) = x(t) \). If \( u(t) = 1 \), the state changes, i.e., \( x(t+1) = 1 - x(t) \). The state transition function can be defined as:\[x(t+1) = x(t) \oplus u(t)\]where \( \oplus \) denotes the XOR operation.

Output Function

The output \( y(t) \) depends on the state \( x(t) \). According to the problem, the output should be 1 if the number of 1's seen so far (excluding current input) is even, which is represented by \( x(t) = 0 \). Therefore, the output function is:\[ y(t) = 1 - x(t) \]

State Space Formulation

We can now define the state space representation of the system. - **State Transition Function**: \( x(t+1) = x(t) \oplus u(t) \)- **Output Function**: \( y(t) = 1 - x(t) \)This provides a complete description in state space form where the state \( x(t) \) keeps track of the parity of the number of 1's seen so far and the output \( y(t) \) is determined based on this state.

Key concepts

Discrete-Time Systems

Discrete-time systems are a fundamental concept in control theory and signal processing. Unlike continuous-time systems, which deal with signals that are continuous over time, discrete-time systems focus on signals and operations that occur at distinct and separate time steps. This means that the system undergoes state changes at specific intervals, which are usually evenly spaced in time.

These systems are particularly suitable for digital computers and electronic systems because they operate using sequential data inputs. Discrete-time systems form the basis for digital signal processing and are instrumental in various applications, such as control systems, telecommunications, and data compression. By understanding discrete-time systems, you can model and analyze systems where actions occur at discrete time intervals, which is the case for many real-world applications.

State Transition Function

The state transition function is a key component in the representation of a state-space model. It describes how the system transitions from one state to another based on the current state and input. For a discrete-time system, this function is defined mathematically, capturing the dynamics and behavior of the system at each time step.

In the context of our example, the state transition function is:

• Function: \( x(t+1) = x(t) \oplus u(t) \)
• Here, \( x(t) \) is the current state representing the parity of 1's seen so far,
• \( u(t) \) is the current input, and
• \( \oplus \) denotes the XOR operation.

This equation means that the state will toggle between 0 and 1 when the input is 1, indicating a change in parity, and remain unchanged when the input is 0. This function is vital for tracking the system's state accurately over time.

Output Function

The output function defines how the system's output at any given time relates to its current state. It is responsible for producing the observable output based on the internal state of the system.

In our exercise, the output function can be represented as:

• Function: \( y(t) = 1 - x(t) \)
• This means that the output \( y(t) \) is 1 if the state \( x(t) \) is 0, i.e., when there is an even number of 1's in the input stream so far (excluding the current input).
• On the other hand, the output is 0 if the state is 1, indicating an odd parity.

This simple yet effective function ensures that the system outputs a signal reflecting the parity of the 1's counted, thus aligning with the problem's requirements.

Parity Checking

Parity checking is a critical process in error detection, ensuring that data integrity is maintained during transmission or storage. It involves counting the number of '1's in a binary input and determining whether this count is even or odd.

In digital systems, parity checking can quickly flag errors by comparing the received data's parity to an expected value, thus revealing inconsistencies. In this exercise, parity checking is achieved by monitoring the state changes defined by the state transition function, which keeps track of the number of 1's seen.

This approach allows the system to detect if the input sequence has an even number of ones, enabling the output function to produce the appropriate signal. By leveraging state-space representation, parity checking becomes an efficient tool for verifying data consistency in discrete-time systems.