Linear Time Invariant System and its Analysis

Contents

**Describe LTI System and its Block diagram** 2

**Describe Continuous-Time and Discrete-Time LTI Systems** 3

**Describe Impulse response of Continuous-Time and Discrete-Time LTI Systems** 4

**Describe Discrete-Time Convolution and its properties** 5

**Explain graphical method for Discrete-Time Convolution** 6

**Explain the Matrix method for Discrete-Time Convolution** 7

**Explain the general formula for Discrete-Time Convolution** 8

**Explain Discrete-Time LTI System described by the Difference equation** 9

**Describe Continuous-Time Convolution and its properties** 10

**Explain General formula for Continuous-Time Convolution** 11

**Explain Graphical method for Continuous-Time Convolution** 12

**Explain Continuous-Time LTI System described by Differential equation** 12

**Describe Eigen functions of Continuous-Time System and Discrete-Time System** 13

**Explain the Cascade connection and Parallel connection of LTI System** 14

**Describe Causal and Non-Causal Discrete-Time and Continuous-Time LTI systems** 15

**Describe Stable and Unstable Discrete-Time and Continuous-Time LTI systems** 16

**Describe Stable and Unstable Discrete-Time and Continuous-Time LTI systems** 18

**Describe LTI System and its Block diagram**

LTI System stands for Linear Time-Invariant System. It is a type of system in which the output is a linear transformation of the input and the system characteristics remain constant over time.

In an LTI system, the output is a weighted sum of the input signal and its delayed versions. The weights or coefficients determine how the input signal is processed or transformed by the system. The system is time-invariant because the transformation applied to the input signal remains the same regardless of when the input is applied.

The block diagram is a graphical representation of an LTI system. It consists of blocks that represent the individual components or operations of the system, and arrows that represent the flow of signals between these components. Each block performs a specific operation on the input signal to produce the output signal.

The basic blocks used in an LTI system block diagram include:

- Input block: Represents the input signal to the system.
- System block: Represents the LTI system itself. It may include operations such as filtering, amplification, modulation, etc.
- Output block: Represents the output signal of the system.
- Summing block: Represents the addition or subtraction operation, where multiple signals are combined or subtracted.
- Delay block: Represents a time delay applied to the signal.

The block diagram provides a visual representation of how the input signal is processed through the system to generate the output signal. It allows for a clear understanding of the flow of signals and the sequence of operations performed by the system.

By analyzing the block diagram, one can determine the overall transfer function of the system, which describes the relationship between the input and output signals in the frequency domain. This transfer function can be used to analyze the system’s response to different input signals and to design and optimize the system for desired performance.

**Describe Continuous-Time and Discrete-Time LTI Systems**

Continuous-Time LTI Systems:

A continuous-time LTI (Linear Time-Invariant) system operates on continuous-time signals. The input and output signals of the system are continuous functions of time. These systems are characterized by the properties of linearity and time-invariance. The mathematical representation of a continuous-time LTI system is often done using differential equations or frequency response functions.

The input-output relationship of a continuous-time LTI system can be described by a convolution integral, which relates the input signal x(t) to the output signal y(t) as follows:

y(t) = x(t) * h(t)

where * denotes the convolution operation and h(t) is the system’s impulse response. The impulse response represents the system’s behavior when the input is an impulse function (δ(t)).

Discrete-Time LTI Systems:

A discrete-time LTI system operates on discrete-time signals. The input and output signals of the system are sequences of numbers, usually sampled at regular intervals. These systems are characterized by the properties of linearity and time-invariance, similar to continuous-time LTI systems.

The input-output relationship of a discrete-time LTI system can be described by a convolution sum, which relates the input sequence x[n] to the output sequence y[n] as follows:

y[n] = x[n] * h[n]

where * denotes the convolution operation and h[n] is the system’s impulse response. The impulse response represents the system’s behavior when the input is an impulse sequence (δ[n]).

In both continuous-time and discrete-time LTI systems, the linearity property means that the system responds proportionally to the input signal, and the time-invariance property means that the system’s behavior remains constant over time.

The analysis and design of continuous-time and discrete-time LTI systems involve techniques such as Fourier analysis, Laplace transforms, Z-transforms, and frequency response analysis. These tools help understand the system’s characteristics, stability, frequency response, and other performance parameters.

**Describe Impulse response of Continuous-Time and Discrete-Time LTI Systems**

Impulse response is a fundamental concept in the analysis and characterization of linear time-invariant (LTI) systems, both in continuous-time and discrete-time domains.

Continuous-Time LTI Systems:

In a continuous-time LTI system, the impulse response represents the system’s output when the input is an impulse function, typically denoted as δ(t). Mathematically, the impulse response is denoted as h(t). It describes the system’s behavior and the relationship between the input and output signals over time.

The convolution integral expresses the output of a continuous-time LTI system as the convolution of the input signal x(t) with the impulse response h(t):

y(t) = x(t) * h(t)

The impulse response provides information about how the system responds to different input signals. By convolving the input signal with the impulse response, the output of the system for any arbitrary input can be determined.

Discrete-Time LTI Systems:

In a discrete-time LTI system, the impulse response represents the system’s output when the input is an impulse sequence, typically denoted as δ[n]. Mathematically, the impulse response is denoted as h[n]. It describes the system’s behavior and the relationship between the input and output sequences.

The convolution sum expresses the output of a discrete-time LTI system as the convolution of the input sequence x[n] with the impulse response h[n]:

y[n] = x[n] * h[n]

Similar to the continuous-time case, the impulse response provides information about the system’s response to different input sequences. By convolving the input sequence with the impulse response, the output of the system for any arbitrary input can be determined.

The impulse response of an LTI system characterizes its unique properties, such as stability, frequency response, and time-domain behavior. It allows for the analysis and understanding of how the system modifies and shapes the input signal to produce the corresponding output signal.

**Describe Discrete-Time Convolution and its properties**

Discrete-Time Convolution is a mathematical operation that is used to calculate the output of a discrete-time linear time-invariant (LTI) system for any given input. The output of a discrete-time LTI system can be obtained by convolving the input sequence with the impulse response of the system. The convolution operation is denoted by the symbol “*”.

If x[n] is the input sequence and h[n] is the impulse response of the system, then the output sequence y[n] can be obtained as:

y[n] = x[n] * h[n] = ∑x[k]h[n-k], where the summation is taken over all possible values of k for which x[k] and h[n-k] are defined.

The properties of discrete-time convolution are as follows:

- Commutativity: The order of convolution does not affect the result, i.e., x[n] * h[n] = h[n] * x[n].
- Associativity: The order of grouping of the sequences does not affect the result, i.e., (x[n] * h1[n]) * h2[n] = x[n] * (h1[n] * h2[n]).
- Distributivity: Convolution distributes over addition, i.e., x[n] * (h1[n] + h2[n]) = x[n] * h1[n] + x[n] * h2[n].
- Linearity: Convolution is a linear operation, i.e., a1 x1[n] + a2 x2[n] * h[n] = a1 x1[n] * h[n] + a2 x2[n] * h[n].
- Time-invariance: If the input sequence is shifted by k samples, then the output sequence is also shifted by k samples, i.e., x[n-k] * h[n] = x[n] * h[n-k].
- Associativity with scaling: Convolution is associative with scaling, i.e., a(x[n] * h[n]) = (axe[n]) * h[n] = x[n] * (ah[n]).

These properties are very useful in the analysis and design of digital signal processing systems.

**Explain graphical method for Discrete-Time Convolution**

The graphical method for discrete-time convolution involves plotting the signals being convolved on a graph and then sliding one signal over the other while calculating the area of overlap at each point. The result of convolution is the sum of the areas of overlap at each point.

To perform discrete-time convolution using the graphical method, follow these steps:

- Plot the two signals on a graph, with one signal on the x-axis and the other signal on the y-axis.
- Flip one of the signals (usually the impulse response) about the origin.
- Slide the flipped signal along the x-axis and calculate the area of overlap at each point.
- Record the area of overlap at each point.
- Add up all the recorded areas of overlap to obtain the output signal.

The properties of discrete-time convolution include:

- Commutativity: Changing the order of the signals being convolved does not affect the result.
- Associativity: Changing the grouping of the signals being convolved does not affect the result.
- Distributivity: Convolution is distributive over addition.
- Time invariance: If a system is time-invariant, then convolution of the input signal with the impulse response will produce the same output signal, regardless of when the input signal is applied.

**Explain the Matrix method for Discrete-Time Convolution**

The matrix method for discrete-time convolution is a simple and efficient way to perform the convolution of two discrete-time signals. It involves representing the two signals as matrices and multiplying them together using matrix multiplication.

Consider two discrete-time signals x[n] and h[n] of lengths N and M respectively. We can represent these signals as matrices X and H of dimensions N x 1 and M x 1 respectively, as follows:

X = [x[0] x[1] x[2] … x[N-1]]^{T}

H = [h[0] h[1] h[2] … h[M-1]]^{T}

To perform the convolution of x[n] and h[n], we first create a matrix H’ by appending M-1 zeros to the end of H:

H’ = [h[0] h[1] h[2] … h[M-1] 0 0 0 … 0]^{T}

The convolution of x[n] and h[n] can then be obtained as the product of the convolution matrix C and the matrix X, where C is a Toeplitz matrix obtained by convolving H’ with a shifted version of itself:

C = [h[0] 0 0 … 0 0 0]

[0 h[M-1] h[M-2] … h[2] h[1] h[0]]

The product of C and X gives us the convolution of x[n] and h[n] as a matrix of length N+M-1:

Y = C*X

The resulting matrix Y can be truncated to obtain the first N+M-1 samples of the convolution of x[n] and h[n].

**Explain the general formula for Discrete-Time Convolution**

The general formula for discrete-time convolution relates the input sequence, the impulse response, and the output sequence of a discrete-time linear time-invariant (LTI) system. The formula is as follows:

y[n] = ∑[k=-∞ to ∞] x[k] * h[n – k]

Where:

- y[n] represents the output sequence at time index n.
- x[k] represents the input sequence at time index k.
- h[n – k] represents the impulse response sequence, delayed by k samples, at time index n.

In this formula, the input sequence x[k] and the impulse response sequence h[n – k] are multiplied at each time index k, and the results are summed over all possible values of k. The resulting summation gives the value of the output sequence y[n] at each time index n.

The discrete-time convolution operation can be interpreted as sliding the impulse response sequence over the input sequence, multiplying corresponding samples, and summing the products to obtain the output sequence. It captures the way the LTI system responds to different inputs and produces the corresponding outputs.

The general formula for discrete-time convolution is an essential tool in analyzing and understanding the behavior of discrete-time systems, as it allows for the computation of the output sequence based on the input sequence and the impulse response.

**Explain Discrete-Time LTI System described by the Difference equation**

A discrete-time LTI system can be described using a difference equation. The difference equation relates the output of the system to the current and past inputs and outputs of the system.

The general form of a first-order difference equation is:

y[n] = a y[n-1] + b x[n]

where y[n] is the output of the system at time n, x[n] is the input of the system at time n, and a and b are constants.

For a second-order difference equation, the general form is:

y[n] = a1 y[n-1] + a2 y[n-2] + b0 x[n] + b1 x[n-1] + b2 x[n-2]

where y[n] is the output of the system at time n, x[n] is the input of the system at time n, and a1, a2, b0, b1, and b2 are constants.

The impulse response of a discrete-time LTI system described by a difference equation can be found by applying an impulse input (i.e., a Kronecker delta) to the system and solving the resulting difference equation.

Once the impulse response is found, the output of the system can be calculated for any input signal by convolving the input signal with the impulse response.

**Describe Continuous-Time Convolution and its properties**

Continuous-Time Convolution is a mathematical operation used to determine the output of a linear time-invariant (LTI) system when given an input signal. The convolution operation involves integrating the product of the input signal and the impulse response of the LTI system over time. The resulting output signal represents the system’s response to the input signal.

Mathematically, the continuous-time convolution operation can be expressed as:

y(t) = ∫[h(τ) x(t-τ)]dτ

where y(t) is the output signal, x(t) is the input signal, h(t) is the impulse response of the system, and the integral is taken over all time values.

Properties of Continuous-Time Convolution:

- Commutative property: x(t) * h(t) = h(t) * x(t)
- Associative property: [x(t) * h1(t)] * h2(t) = x(t) * [h1(t) * h2(t)]
- Distributive property: x(t) * [h1(t) + h2(t)] = x(t) * h1(t) + x(t) * h2(t)
- Convolution with the impulse response of a system gives the system’s response to any input signal.
- Convolution in time domain corresponds to multiplication in frequency domain, i.e., X(f) H(f) = Y(f), where X(f), H(f), and Y(f) are the Fourier transforms of x(t), h(t), and y(t), respectively.
- Convolution in time domain with a unit step function gives the integral of the function, i.e., x(t) * u(t) = ∫x(τ) dτ, where u(t) is the unit step function.

Continuous-Time Convolution is a fundamental operation in signal processing and is used in many applications, including image processing, audio processing, and communication systems.

**Explain General formula for Continuous-Time Convolution**

The general formula for Continuous-Time Convolution involves the integration of the product of two continuous-time functions, which can be expressed as follows:

y(t) = ∫[x(τ)h(t-τ)] dτ

Where, x(t) and h(t) are the input and impulse response functions, respectively, and y(t) is the output function resulting from the convolution operation.

This formula represents the mathematical operation used to determine the output of a linear time-invariant (LTI) system when given an input signal. It involves integrating the product of the input signal and the impulse response of the LTI system over time, resulting in the system’s response to the input signal.

The limits of the integral depend on the functions being convolved. For example, if both x(t) and h(t) are finite-duration signals, the limits of the integral would be from -∞ to +∞, since the product of the two functions is non-zero only over the range where both are non-zero. On the other hand, if either x(t) or h(t) is an infinite-duration signal, the limits of the integral would be determined accordingly.

**Explain Graphical method for Continuous-Time Convolution**

The graphical method for Continuous-Time Convolution is a way to compute the output of a linear time-invariant (LTI) system when given an input signal and its impulse response graphically. It involves plotting the input signal and impulse response on a graph and sliding one of them across the other while multiplying their overlapping areas.

The graphical method can be summarised in the following steps:

- Plot the input signal x(t) and impulse response h(t) on a graph, with x-axis representing time and y-axis representing amplitude.
- Flip the impulse response h(t) about the y-axis to obtain h(-t), which represents the time-reversed impulse response.
- Slide the flipped impulse response h(-t) along the time axis of the input signal x(t), with its right end touching the left end of x(t).
- For each value of t, multiply the overlapping area of x(t) and h(-t) to obtain the value of the output signal y(t).
- Plot the resulting output signal y(t) on the same graph as the input signal and impulse response.
- Repeat the process for each value of t, sliding the flipped impulse response along the time axis of the input signal.
- The resulting plot of the output signal y(t) is the convolution of the input signal x(t) and impulse response h(t).

**Explain Continuous-Time LTI System described by Differential equation**

A continuous-time linear time-invariant (LTI) system can be described by a differential equation. The differential equation represents the relationship between the input signal and the output signal of the system.

The general form of the differential equation for a continuous-time LTI system is as follows:

aₙ dⁿy(t)/dtⁿ + aₙ₋₁ dⁿ⁻¹y(t)/dtⁿ⁻¹ + … + a₁ dy(t)/dt + a₀ y(t) = bₘ dᵐx(t)/dtᵐ + bₘ₋₁ dᵐ⁻¹x(t)/dtᵐ⁻¹ + … + b₁ dx(t)/dt + b₀ x(t)

Where:

- y(t) is the output signal of the system as a function of time.
- x(t) is the input signal of the system as a function of time.
- aₙ, aₙ₋₁, …, a₁, a₀ are the coefficients of the derivatives of the output signal y(t).
- bₘ, bₘ₋₁, …, b₁, b₀ are the coefficients of the derivatives of the input signal x(t).
- dⁿy(t)/dtⁿ represents the nth derivative of the output signal y(t) with respect to time.
- dᵐx(t)/dtᵐ represents the mth derivative of the input signal x(t) with respect to time.

The differential equation describes how the output signal y(t) of the system changes over time based on the input signal x(t) and the coefficients of the derivatives. It captures the dynamic behavior of the continuous-time LTI system.

Solving the differential equation allows us to determine the relationship between the input and output signals and analyze the system’s response to different inputs. Techniques such as Laplace transforms, Fourier transforms, and numerical methods can be employed to solve the differential equation and obtain the output signal for a given input signal.

**Describe Eigen functions of Continuous-Time System and Discrete-Time System**

Eigenfunctions are a fundamental concept in the analysis of continuous-time and discrete-time systems. Let’s understand eigenfunctions in both contexts:

- Eigenfunctions of Continuous-Time Systems:

In continuous-time systems, an eigenfunction is a special type of input signal that produces a scaled version of itself as the output. Mathematically, if x(t) is an eigenfunction of a continuous-time system with an associated eigenvalue λ, then the system response is given by y(t) = H{x(t)} = λx(t), where H{} represents the system operation.

Eigenfunctions play a crucial role in the analysis of continuous-time systems. They help in understanding the system’s behavior, stability, and response to different inputs. Common examples of eigenfunctions in continuous-time systems include sinusoidal signals (e.g., sine and cosine functions) and complex exponential signals.

- Eigenfunctions of Discrete-Time Systems:

In discrete-time systems, eigenfunctions are also input signals that produce a scaled version of themselves as the output. If x[n] is an eigenfunction of a discrete-time system with an associated eigenvalue λ, then the system response is given by y[n] = H{x[n]} = λx[n].

Similar to continuous-time systems, eigenfunctions in discrete-time systems are useful in analyzing system behavior and properties. Some examples of eigenfunctions in discrete-time systems include discrete complex exponential sequences, unit impulse sequences, and unit step sequences.

Eigenfunctions provide insights into the system’s characteristics and can be used to determine important system properties such as stability, frequency response, and impulse response.

In summary, eigenfunctions in both continuous-time and discrete-time systems are input signals that result in a scaled version of themselves as the system output. They are essential tools for analyzing and understanding the behavior of systems in the respective domains.

**Explain the Cascade connection and Parallel connection of LTI System**

Cascade Connection of LTI Systems:

In cascade connection, two or more LTI (Linear Time-Invariant) systems are connected in series, where the output of one system serves as the input to the next system. The overall transfer function of the cascade connection is obtained by multiplying the individual transfer functions of the systems.

Let’s consider two LTI systems, System 1 and System 2, with transfer functions H1(s) and H2(s), respectively. The cascade connection of these systems can be represented as:

Input —-> [System 1] —-> [System 2] —-> Output

The transfer function of the cascade connection is given by:

H(s) = H1(s) * H2(s)

The advantage of the cascade connection is that it allows us to break down a complex system into simpler subsystems. Each subsystem can be designed and analyzed separately, and the overall system response is obtained by cascading the individual subsystem responses.

Parallel Connection of LTI Systems:

In parallel connection, two or more LTI systems are connected in parallel, where the same input signal is applied to all systems simultaneously, and their individual outputs are summed to obtain the overall output. The transfer function of the parallel connection is obtained by adding the individual transfer functions of the systems.

Let’s consider two LTI systems, System 1 and System 2, with transfer functions H1(s) and H2(s), respectively. The parallel connection of these systems can be represented as:

+—- [System 1] —-+

Input —-> —–| |—-> Output

+—- [System 2] —-+

The transfer function of the parallel connection is given by:

H(s) = H1(s) + H2(s)

The advantage of the parallel connection is that it allows us to combine the outputs of multiple systems to achieve a desired overall response.

This is commonly used in systems where different subsystems contribute to different aspects of the overall system behavior.

Both cascade and parallel connections are widely used in the analysis and design of LTI systems to achieve desired system responses and characteristics.

**Describe Causal and Non-Causal Discrete-Time and Continuous-Time LTI systems**

In signal processing, a system is said to be causal if the output of the system at any given time depends only on past and current inputs, and not on any future inputs. A non-causal system, on the other hand, can produce output that depends on future inputs.

Causal Continuous-Time LTI Systems: A continuous-time LTI system is causal if its impulse response is zero for negative time. That is, if h(t) = 0 for t < 0. Causal systems are often used in real-time applications, where the output must depend only on past and current inputs. For example, a simple low-pass filter can be a causal continuous-time LTI system.

Non-Causal Continuous-Time LTI Systems: A continuous-time LTI system is non-causal if its impulse response is non-zero for negative time. That is, if h(t) ≠ 0 for t < 0. Non-causal systems are often used in situations where future inputs can affect the output. For example, a predictive filter that uses future samples to estimate the current output can be a non-causal continuous-time LTI system.

Causal Discrete-Time LTI Systems: A discrete-time LTI system is causal if its impulse response is zero for negative time index. That is, if h[n] = 0 for n < 0. Causal systems are often used in real-time digital signal processing applications, where the output must depend only on past and current inputs. For example, a simple moving average filter can be a causal discrete-time LTI system.

Non-Causal Discrete-Time LTI Systems: A discrete-time LTI system is non-causal if its impulse response is non-zero for negative time index. That is, if h[n] ≠ 0 for n < 0. Non-causal systems are often used in situations where future inputs can affect the output. For example, a predictive filter that uses future samples to estimate the current output can be a non-causal discrete-time LTI system.

**Describe Stable and Unstable Discrete-Time and Continuous-Time LTI systems**

Stable LTI Systems:

A stable LTI (Linear Time-Invariant) system is one in which bounded input signals result in bounded output signals. In other words, if the input to the system is finite and bounded, then the output of a stable LTI system will also be finite and bounded. Stability is an essential property of systems to ensure their reliable and predictable behavior.

Discrete-Time LTI Systems:

In the context of discrete-time LTI systems, stability is determined by the region of convergence (ROC) of the system’s transfer function. If the ROC includes the unit circle in the complex plane, then the system is considered stable. This means that the poles of the transfer function should lie inside the unit circle for stability.

Continuous-Time LTI Systems:

For continuous-time LTI systems, stability is determined by the eigenvalues of the system’s characteristic equation or the poles of the transfer function. If all the poles have negative real parts, then the system is stable. This means that the exponential components of the output decay over time, leading to a bounded output for bounded inputs.

Unstable LTI Systems:

An unstable LTI system is one in which bounded input signals result in unbounded or diverging output signals. In other words, if the input to the system is finite and bounded, the output of an unstable LTI system will grow infinitely over time or exhibit oscillatory behavior. Unstable systems are undesirable as they do not provide reliable and predictable responses.

Discrete-Time LTI Systems:

In discrete-time LTI systems, instability occurs when the ROC of the transfer function does not include the unit circle. This typically happens when the poles of the transfer function are outside the unit circle, leading to unbounded or oscillatory output responses.

Continuous-Time LTI Systems:

For continuous-time LTI systems, instability occurs when one or more poles of the transfer function have positive real parts. This leads to exponential growth or oscillatory behavior in the output response.

It is important to ensure stability in LTI systems to avoid unpredictable and uncontrollable behavior. Stable systems are capable of providing consistent and reliable responses to input signals, making them suitable for various applications in signal processing, control systems, and communication systems.

**Describe Stable and Unstable Discrete-Time and Continuous-Time LTI systems**

Stable LTI Systems:

A stable LTI (Linear Time-Invariant) system is one in which bounded input signals result in bounded output signals. In other words, if the input to the system is finite and bounded, then the output of a stable LTI system will also be finite and bounded. Stability is an essential property of systems to ensure their reliable and predictable behavior.

Discrete-Time LTI Systems:

In the context of discrete-time LTI systems, stability is determined by the region of convergence (ROC) of the system’s transfer function. If the ROC includes the unit circle in the complex plane, then the system is considered stable. This means that the poles of the transfer function should lie inside the unit circle for stability.

Continuous-Time LTI Systems:

For continuous-time LTI systems, stability is determined by the eigenvalues of the system’s characteristic equation or the poles of the transfer function. If all the poles have negative real parts, then the system is stable. This means that the exponential components of the output decay over time, leading to a bounded output for bounded inputs.

Unstable LTI Systems:

An unstable LTI system is one in which bounded input signals result in unbounded or diverging output signals. In other words, if the input to the system is finite and bounded, the output of an unstable LTI system will grow infinitely over time or exhibit oscillatory behavior. Unstable systems are undesirable as they do not provide reliable and predictable responses.

Discrete-Time LTI Systems:

In discrete-time LTI systems, instability occurs when the ROC of the transfer function does not include the unit circle. This typically happens when the poles of the transfer function are outside the unit circle, leading to unbounded or oscillatory output responses.

Continuous-Time LTI Systems:

For continuous-time LTI systems, instability occurs when one or more poles of the transfer function have positive real parts. This leads to exponential growth or oscillatory behavior in the output response.

It is important to ensure stability in LTI systems to avoid unpredictable and uncontrollable behavior. Stable systems are capable of providing consistent and reliable responses to input signals, making them suitable for various applications in signal processing, control systems, and communication systems.