All questions of Introduction to Signals & Systems for Electrical Engineering (EE) Exam

The period of a sinusoidal signal of the form x(t) = A*sin(ωt + φ) is given by T = 2π/ω, where ω is the angular frequency. In this case, ω = 18π, so the period is T = 2π/(18π) = 1/9 seconds. The phase angle (in degrees) does not affect the period. Therefore, the period of the given signal is 1/9 seconds.

The function obeys the scaling/homogeneity property, but doesn’t obey the additivity property, thus not being linear.

Sin x is a periodic function, but log x is not a periodic function. Thus y is log t, where t= sin x, thus y is not periodic.

Correct option is not C.
first t is missing
if t is there fundamental frequency would be 2pi*w

Reduces to 1 – 2t, which is a strictly decreasing function.

Understanding the Sinusoidal Signal
The given sinusoidal signal is:
- x(t) = 2sin(100t + π/3)
This signal oscillates between -2 and +2, with a frequency determined by the coefficient of t.
Passing through the Square Law Device
The square law device has the relationship:
- y(t) = x^2(t)
To find the output signal, we need to compute x^2(t):
- y(t) = (2sin(100t + π/3))^2
- y(t) = 4sin^2(100t + π/3)
Using Trigonometric Identity
We can simplify sin^2 using the identity:
- sin^2(θ) = (1 - cos(2θ))/2
Applying this to our equation:
- y(t) = 4 * (1 - cos(200t + 2π/3))/2
- y(t) = 2(1 - cos(200t + 2π/3))
- y(t) = 2 - 2cos(200t + 2π/3)
Identifying the DC Component
The DC component of a signal is defined as the constant term (the part that does not oscillate). From the expression of y(t):
- The term "2" is the DC component.
- The term "-2cos(200t + 2π/3)" oscillates and does not contribute to the DC component.
Conclusion
Thus, the DC component in the signal y(t) is:
- 2
The correct answer is option 'B'.

Explanation:

The given equation is Y(t) = x(t/5), which implies that the output signal Y(t) is obtained by compressing the input signal x(t) by a factor of 5. This means that the signal is scaled down in time, and the time axis is compressed.

Therefore, the correct option is (B) Expanded signal.

Here are some key points to understand:

- The input signal x(t) is compressed by a factor of 5. This means that the time axis is scaled down by a factor of 5.
- The output signal Y(t) is obtained by applying this compression to the input signal.
- Since the time axis is compressed, the signal appears to expand in amplitude. This is because the same amount of signal is now squeezed into a smaller time interval.
- Therefore, the output signal Y(t) is an expanded signal, as compared to the input signal x(t).

In summary, the given equation Y(t) = x(t/5) represents an expanded signal, as the input signal is compressed in time to obtain the output signal.

The System Transfer Function and its Input

The system transfer function is a mathematical representation of a linear, time-invariant system. It relates the input to the output of the system and provides valuable information about the system's behavior and response. The transfer function is typically represented by a ratio of polynomials in the Laplace domain.

Definition of Transfer Function
The transfer function of a system is defined as the Laplace transform of the system's output divided by the Laplace transform of its input, assuming zero initial conditions. Mathematically, it can be represented as:

H(s) = Y(s) / X(s)

where H(s) is the transfer function, Y(s) is the Laplace transform of the system's output, and X(s) is the Laplace transform of the system's input.

Exchanging the Transfer Function and Input
The given statement states that if we exchange the transfer function and the input, we will still obtain the same response. In other words, if we interchange the roles of the input and output in the transfer function, the resulting transfer function will yield the same output for a given input.

Explanation of True Answer
This statement is true because the system transfer function is a mathematical representation of the system's behavior and response, which is independent of the specific input signal. The transfer function captures the system's dynamics, such as the relationship between the input and output signals, the system's poles and zeros, and the frequency response.

When the input and output are interchanged, the resulting transfer function will still capture the same system dynamics. The only difference is that the roles of the input and output signals are reversed. However, the system's behavior and response remain the same.

Example:
For example, consider a simple system with a transfer function H(s) = 1/s. If we apply an input signal X(s) = 1/s^2, the output Y(s) can be calculated by multiplying the transfer function by the input:

Y(s) = H(s) * X(s) = (1/s) * (1/s^2) = 1/s^3

If we interchange the roles of the input and output signals, the resulting transfer function becomes H'(s) = 1/s^3. Now, if we apply an input signal X'(s) = 1/s^2 to this new transfer function, the output Y'(s) can be calculated as:

Y'(s) = H'(s) * X'(s) = (1/s^3) * (1/s^2) = 1/s^5

As we can see, even though the input and transfer function have been exchanged, the response of the system remains the same. This example illustrates the validity of the given statement and why it is true.

Conclusion
In conclusion, the system transfer function and the input can be interchanged without affecting the system's response. The transfer function captures the system's dynamics, which are independent of the specific input signal. Exchanging the input and transfer function simply reverses the roles of the input and output signals, but the system's behavior and response remain the same.

Inverse System:

The given system is y(t) = 2x(t). We need to find the inverse system of this given system.

The inverse system is obtained by swapping the input and output of the given system. Therefore, the output of the inverse system becomes the input of the given system, and vice versa.

To find the inverse system, we can express the given system as y(t) = H[x(t)], where H represents the system or operator.

Now, let's swap the input and output variables:

x(t) = H^(-1)[y(t)]

This equation represents the inverse system, where x(t) is the output of the inverse system and y(t) is the input.

Comparing this equation with the given options, we can see that option A satisfies the equation. Therefore, the inverse system is given by y(t) = 0.5x(t).

Explanation of Option A:

Option A states that the output of the inverse system, y(t), is equal to 0.5 times the input of the inverse system, x(t).

This means that for a given input x(t), the output of the inverse system will be half of the input. In other words, the inverse system attenuates the input signal by a factor of 0.5.

If we apply this inverse system to the output of the given system y(t) = 2x(t), we should obtain the original input signal x(t) as the output.

Let's verify this:

Given system: y(t) = 2x(t)

Inverse system (Option A): y(t) = 0.5x(t)

If we apply the given system followed by the inverse system, we get:

y(t) = 2x(t) (Given system)
y(t) = 0.5(2x(t)) (Inverse system)
y(t) = x(t)

As we can see, the output of the inverse system is equal to the original input signal x(t). Hence, option A represents the correct inverse system.

Final Answer: Option A

Explanation: The fundamental period = 2pi/(pi/4) = 8.

The continuous-time impulse function, denoted as δ(t), is a fundamental concept in signal processing and systems theory. It is often used to represent an infinitesimally narrow pulse or spike in a continuous-time signal. The impulse function can be defined in terms of the step function, denoted as u(t), as follows:



Explanation:
The step function u(t) is defined as:

u(t) =
1, t >= 0
0, t < />

The derivative of the step function u(t) with respect to time, du/dt, can be calculated as:

du/dt =
0, t < />
∞, t = 0
0, t > 0

The derivative of the step function u(t) is zero for t < 0,="" remains="" undefined="" at="" t="0," and="" becomes="" infinite="" for="" t="" /> 0. This behavior is characteristic of the impulse function.

Now, let's consider the derivative of the impulse function δ(t), denoted as d(t). By definition, the impulse function δ(t) is the derivative of the step function u(t):

d(t) = du/dt

So, the correct answer is option 'C': d(t) = du/dt.

Compressed signal

Explanation:
- The given signal is Y(t) = x(2t), which means the input signal x(t) is being multiplied by a factor of 2 in the argument of t.
- This leads to a compression of the signal in the time domain. This is because the same input signal is being represented in a shorter time duration.
- To understand this, let us consider an example. Suppose x(t) is a signal that represents a sine wave with a period of T seconds. If we apply the transformation Y(t) = x(2t), then the resulting signal will have a period of T/2 seconds. This is because the time axis has been compressed by a factor of 2.
- Therefore, we can conclude that the given signal Y(t) = x(2t) is a compressed signal.

To find the amplitude and frequency components of the given signal x(t), we can break it down into its cosine and sine components separately:

x(t) = 4 cos (2t/3) + 8 sin (0.5t) + 7 sin (t/3)

The amplitude of a cosine or sine function can be found by taking the coefficient in front of it.

For the cosine component:
Amplitude = 4

For the sine components:
Amplitude = 8 for sin (0.5t)
Amplitude = 7 for sin (t/3)

The frequency of a cosine or sine function can be found by taking the coefficient of t inside the function and dividing it by 2π.

For the cosine component:
Frequency = 2/3π

For the sine components:
Frequency = 0.5/2π = 1/4π for sin (0.5t)
Frequency = 1/3π for sin (t/3)

Therefore, the amplitude and frequency components of the given signal x(t) are as follows:

Amplitude:
Cosine component = 4
Sine component 1 = 8
Sine component 2 = 7

Frequency:
Cosine component = 2/3π
Sine component 1 = 1/4π
Sine component 2 = 1/3π

Sorry, but I need more information in order to provide a response.

Integrator System Stability

An integrator system is a type of system that performs mathematical integration of the input signal. It is commonly used in control systems, signal processing, and other applications where the integration of a signal is required. The stability of a system refers to its ability to produce a bounded output for a bounded input.

To determine the stability of an integrator system, we need to analyze its transfer function. The transfer function of an integrator system can be represented as:

H(s) = 1/s

Where H(s) is the transfer function and s is the Laplace variable.

Bounded Input Bounded Output (BIBO) Stability

One way to determine the stability of a system is through the concept of BIBO stability. A system is said to be BIBO stable if every bounded input produces a bounded output.

In the case of an integrator system, the transfer function has a pole at s=0. A pole at the origin indicates that the system has infinite gain at DC (low-frequency) inputs.

Unstable Nature of Integrator System

Due to the pole at the origin, the integrator system is not BIBO stable. This means that even for a bounded input signal, the output of the integrator system will not be bounded.

For example, if we apply a step input signal to the integrator system, the output will continue to increase indefinitely over time. This unbounded behavior indicates instability in the system.

Conclusion

In conclusion, the integrator system is unstable. It fails to meet the criteria of producing a bounded output for a bounded input, which is the definition of BIBO stability. The presence of a pole at the origin in the transfer function causes the output to grow indefinitely for certain input signals.

Amplitude Scaling

Amplitude scaling refers to the process of adjusting the amplitude or strength of a signal. It is commonly used in electronic circuits to control the magnitude of a signal. There are several devices and techniques that can be employed for amplitude scaling, including electronic amplifiers, electronic attenuators, and adders.

Electronic Amplifier

An electronic amplifier is a device that increases the amplitude of a signal. It takes a weak input signal and produces a larger output signal, thereby amplifying the original signal. Amplifiers are widely used in various applications, such as audio systems, telecommunication systems, and instrumentation.

Amplifiers can be categorized into different types based on their operating characteristics, such as voltage amplifiers, current amplifiers, power amplifiers, and operational amplifiers. Each type of amplifier has its own specific purpose and application.

Electronic Attenuator

An electronic attenuator is a device that reduces the amplitude of a signal. It is used to decrease the strength of a signal without introducing significant distortion or noise. Attenuators are commonly used in communication systems, audio equipment, and test and measurement setups.

Attenuators are available in various configurations, including fixed attenuators and variable attenuators. Fixed attenuators provide a fixed level of attenuation, while variable attenuators allow for adjustable attenuation levels. Attenuators are often expressed in terms of decibels (dB), which indicate the amount of attenuation provided.

Amplitude Scaling using Amplifier and Attenuator

The correct answer to the question is option 'C', which states that both an amplifier and an attenuator are examples of amplitude scaling devices. This means that both devices can be used to adjust the amplitude of a signal, albeit in opposite directions.

An amplifier increases the amplitude of a signal, while an attenuator decreases it. Depending on the requirements of a particular application, either an amplifier or an attenuator can be used to achieve the desired amplitude scaling.

Adder

The option 'D', which suggests an adder as an example of amplitude scaling, is incorrect. An adder is a device used in digital circuits to perform addition operations, such as adding two binary numbers. It is not used for amplitude scaling purposes.

In conclusion, amplitude scaling can be achieved using both amplifiers and attenuators. Amplifiers increase the amplitude of a signal, while attenuators decrease it. These devices are widely used in electronic circuits to control and adjust the strength of a signal, depending on the specific requirements of a given application.

Continuous Time Signal vs Discrete Time Signal

Continuous time signals and discrete time signals are two types of signals that are commonly used in the field of signal processing. The main difference between these two types of signals lies in the way they are defined and represented.

Continuous Time Signal:
A continuous time signal x(t) is defined for all values of time t in a continuous manner. This means that the signal is defined at every instant of time in a continuous fashion. The signal can take on any value at any point in time and can be represented by a continuous function.

Examples of continuous time signals include audio signals, video signals, and most real-world signals that are measured or observed in a continuous manner.

Discrete Time Signal:
A discrete time signal x(n) is defined only at specific points in time. The signal is represented by a sequence of values that are sampled at discrete points in time. These discrete points are usually equally spaced and are represented by an index n.

Examples of discrete time signals include digital audio signals, sampled data from sensors, and most signals that are processed by digital systems.

Explanation of the Correct Answer:
The signal denoted by x(t) is known as a continuous time signal. This is because the signal is denoted by a continuous variable "t" and is defined for all values of time in a continuous manner. The signal can take on any value at any point in time, and its representation would require a continuous function.

The other options provided in the question are not correct because:

- Option (a) states that the signal is a discrete time signal, which is not true.
- Option (c) states that the signal is both a discrete time signal and a continuous time signal, which is not correct. The signal cannot be both at the same time.
- Option (d) states that none of the above options are correct, which is also not true.

Therefore, the correct answer is option (b), which states that the signal denoted by x(t) is a continuous time signal.

To express the given finite discrete-time signal as the difference of two unit step sequences, we need to find two unit step sequences that sum up to the given signal.

The unit step sequence u[n] is defined as follows:
u[n] = 1, for n >= 0
u[n] = 0, for n < />

Let's denote the two unit step sequences as u1[n] and u2[n].
u1[n] will represent the step at n = 0, and u2[n] will represent the step at n = N (where N is the last sample of the given signal).

So, u1[n] = u[n] and u2[n] = u[n - N].

Now, let's express the given signal x[n] in terms of the unit step sequences:
x[n] = 1, for 0 <= n=""><=>
0, otherwise

To express x[n] as the difference of two unit step sequences, we can use the following equation:
x[n] = u1[n] - u2[n]

Substituting the expressions for u1[n] and u2[n], we have:
x[n] = u[n] - u[n - N]

Therefore, the given finite discrete-time signal x[n] can be expressed as the difference of two unit step sequences: x[n] = u[n] - u[n - N].

The correct answer is option 'C' - Not always.

Explanation:
A bounded function is a function that has a finite range. In other words, the function does not take on values that go to infinity or negative infinity.

When we talk about the integral of a bounded function from -infinity to infinity, we are referring to the improper integral. The improper integral is defined as the limit of the definite integral as the limits of integration approach infinity and negative infinity.

In some cases, the integral of a bounded function from -infinity to infinity may be defined and finite. This occurs when the function has a well-behaved behavior as the limits of integration go to infinity and negative infinity.

However, there are cases when the integral of a bounded function from -infinity to infinity is not defined or is infinite. This happens when the function has oscillations, or when it approaches infinity or negative infinity as the limits of integration are reached.

For example, consider the function f(x) = sin(x) / x. This function is bounded because the range of sin(x) is between -1 and 1, and dividing by x does not change the boundedness. However, the integral of this function from -infinity to infinity is not defined because it oscillates and does not converge to a finite value.

Therefore, the integral of a bounded function from -infinity to infinity is not always defined and finite. It depends on the behavior of the function as the limits of integration approach infinity and negative infinity.

Integration Operation in Electronic Components

Integration operation in electronic circuits is typically performed by a Capacitor. Let's delve into the details of how a capacitor performs this function:

Capacitor Basics:
- A capacitor is an electronic component that stores electrical energy in an electric field. It consists of two conductive plates separated by an insulating material called a dielectric.

Integration Operation:
- When a capacitor is connected in a circuit, it can accumulate and store electric charge when a voltage is applied across its terminals. This charging and discharging behavior enables the capacitor to perform integration operations in electronic circuits.

Integration Circuit:
- In an integration circuit, a resistor is typically connected in series with a capacitor. When a voltage signal is applied to the input of this circuit, the capacitor starts to accumulate charge based on the input voltage signal. The rate at which the capacitor charges or discharges is determined by the time constant of the circuit, which is a product of the resistance and capacitance values.

Applications:
- Integration circuits are commonly used in signal processing applications such as audio amplifiers, filters, and waveform generators. They can perform mathematical operations like integration to manipulate input signals in various electronic systems.

Conclusion:
- In summary, a capacitor is the component that performs integration operations in electronic circuits by storing and releasing electrical charge based on the input voltage signals. Its ability to accumulate charge over time makes it a crucial element in signal processing and electronic circuit design.