Exponential Function MCQ Quiz - Objective Question with Answer for Exponential Function - Download Free PDF

Explanation:

A signal is a way of conveying information. Gestures, semaphores, images, sound, all can be signals.

Technically - a function of time, space, or another observation variable that conveys information.

• Continuous-Time/Analog Signal
• Discrete-Time Signal
• Digital Signal

Exponential Sequence Graph:

A real exponential signal is defined as:

x(t) = Ae^σt

Where both "A" and "σ" are real. Depending on the value of "σ" the signals will be different. If "σ" is positive the signal x(t) is a growing exponential and if "σ" is negative then the signal x(t) is a decaying exponential. For σ=0, signal x(t) will be constant. 

Hence option(1) represents the exponential signal graph.

Important Points

Option (3) and (4) represents the sinusoidal signal or sine wave.

Sinusoidal Signal:

1. When the Sinewave starts from zero and covers positive values, reaches zero; and again covers negative values, reaches zero, it is said to have completed one cycle or single cycle.
The upper part of the sine wave is called a positive cycle and the lower part is called the negative cycle in a single cycle.

2. Sinusoidal signal need not start at zero. It may start after a certain duration of time. This is time after which Sinusoidal Signal starts is indicated with the help of phase difference (ϕ). It is measured in Radians.

Option (2) represents a discrete-time signal.

A discrete signal or discrete-time signal is a time series consisting of a sequence of quantities. Unlike a continuous-time signal, a discrete-time signal is not a function of a continuous argument; however, it may have been obtained by sampling from a continuous-time signal.

Explanation:

A signal is a way of conveying information. Gestures, semaphores, images, sound, all can be signals.

Technically - a function of time, space, or another observation variable that conveys information.

• Continuous-Time/Analog Signal
• Discrete-Time Signal
• Digital Signal

Exponential Sequence Graph:

A real exponential signal is defined as:

x(t) = Ae^σt

Where both "A" and "σ" are real. Depending on the value of "σ" the signals will be different. If "σ" is positive the signal x(t) is a growing exponential and if "σ" is negative then the signal x(t) is a decaying exponential. For σ=0, signal x(t) will be constant. 

Hence option(1) represents the exponential signal graph.

Important Points

Option (3) and (4) represents the sinusoidal signal or sine wave.

Sinusoidal Signal:

1. When the Sinewave starts from zero and covers positive values, reaches zero; and again covers negative values, reaches zero, it is said to have completed one cycle or single cycle.
The upper part of the sine wave is called a positive cycle and the lower part is called the negative cycle in a single cycle.

2. Sinusoidal signal need not start at zero. It may start after a certain duration of time. This is time after which Sinusoidal Signal starts is indicated with the help of phase difference (ϕ). It is measured in Radians.

Option (2) represents a discrete-time signal.

A discrete signal or discrete-time signal is a time series consisting of a sequence of quantities. Unlike a continuous-time signal, a discrete-time signal is not a function of a continuous argument; however, it may have been obtained by sampling from a continuous-time signal.

Explanation:

A signal is a way of conveying information. Gestures, semaphores, images, sound, all can be signals.

Technically - a function of time, space, or another observation variable that conveys information.

• Continuous-Time/Analog Signal
• Discrete-Time Signal
• Digital Signal

Exponential Sequence Graph:

A real exponential signal is defined as:

x(t) = Ae^σt

Where both "A" and "σ" are real. Depending on the value of "σ" the signals will be different. If "σ" is positive the signal x(t) is a growing exponential and if "σ" is negative then the signal x(t) is a decaying exponential. For σ=0, signal x(t) will be constant. 

Hence option(1) represents the exponential signal graph.

Important Points

Option (3) and (4) represents the sinusoidal signal or sine wave.

Sinusoidal Signal:

1. When the Sinewave starts from zero and covers positive values, reaches zero; and again covers negative values, reaches zero, it is said to have completed one cycle or single cycle.
The upper part of the sine wave is called a positive cycle and the lower part is called the negative cycle in a single cycle.

2. Sinusoidal signal need not start at zero. It may start after a certain duration of time. This is time after which Sinusoidal Signal starts is indicated with the help of phase difference (ϕ). It is measured in Radians.

Option (2) represents a discrete-time signal.

A discrete signal or discrete-time signal is a time series consisting of a sequence of quantities. Unlike a continuous-time signal, a discrete-time signal is not a function of a continuous argument; however, it may have been obtained by sampling from a continuous-time signal.

$\underset{-4}{\overset{4}{\int }}{\mathrm{e}}^{-{\mathrm{t}}^2}\delta \left(2\mathrm{t}-10\right)\mathrm{d}\mathrm{t}=\underset{-4}{\overset{4}{\int }}{\mathrm{e}}^{-{\mathrm{t}}^2}\delta \left[2\left(\mathrm{t}-5\right)\right]\mathrm{d}\mathrm{t}$

$=\underset{-4}{\overset{4}{\int }}\frac{{\mathrm{e}}^{-{\mathrm{t}}^2}}{2}\delta \left(\mathrm{t}-5\right)\mathrm{d}\mathrm{t}$       [using scaling property]

$\Rightarrow \underset{-4}{\overset{4}{\int }}\frac{{\mathrm{e}}^{-{\mathrm{t}}^2}}{2}\delta \left(\mathrm{t}-5\right)\mathrm{d}\mathrm{t}$

$=\underset{-4}{\overset{4}{\int }}\frac{{\mathrm{e}}^{-{\left(5\right)}^2}}{2}\delta \left(\mathrm{t}-5\right)\mathrm{d}\mathrm{t}$     [Using product property]

$=\frac{{\mathrm{e}}^{-25}}{2}\underset{-4}{\overset{4}{\int }}\delta \left(\mathrm{t}-5\right)\mathrm{d}\mathrm{t}=\frac{{\mathrm{e}}^{-25}}{2}.0=0$    [$\delta (\mathrm{t}-5)$ is out of the interval $(-4,4)$]

$\underset{-4}{\overset{4}{\int }}{\mathrm{e}}^{-{\mathrm{t}}^2}\delta \left(2\mathrm{t}-10\right)\mathrm{d}\mathrm{t}=\underset{-4}{\overset{4}{\int }}{\mathrm{e}}^{-{\mathrm{t}}^2}\delta \left[2\left(\mathrm{t}-5\right)\right]\mathrm{d}\mathrm{t}$

$=\underset{-4}{\overset{4}{\int }}\frac{{\mathrm{e}}^{-{\mathrm{t}}^2}}{2}\delta \left(\mathrm{t}-5\right)\mathrm{d}\mathrm{t}$       [using scaling property]

$\Rightarrow \underset{-4}{\overset{4}{\int }}\frac{{\mathrm{e}}^{-{\mathrm{t}}^2}}{2}\delta \left(\mathrm{t}-5\right)\mathrm{d}\mathrm{t}$

$=\underset{-4}{\overset{4}{\int }}\frac{{\mathrm{e}}^{-{\left(5\right)}^2}}{2}\delta \left(\mathrm{t}-5\right)\mathrm{d}\mathrm{t}$     [Using product property]

$=\frac{{\mathrm{e}}^{-25}}{2}\underset{-4}{\overset{4}{\int }}\delta \left(\mathrm{t}-5\right)\mathrm{d}\mathrm{t}=\frac{{\mathrm{e}}^{-25}}{2}.0=0$    [$\delta (\mathrm{t}-5)$ is out of the interval $(-4,4)$]