Complete the sentence correctly: An even signal is _______.
symmetrical about the vertical axis
Even Signal Symmetry Explained
An even signal is a fundamental concept in signal processing and mathematics, characterized by a specific type of symmetry. Understanding the properties of an even signal is crucial for various applications, including Fourier series, Fourier transforms, and general signal analysis.
Defining an Even Signal
Mathematically, a signal \(x(t)\) is classified as an even signal if it satisfies the following condition for all values of \(t\):
\(x(t) = x(-t)\)
This condition means that the value of the signal at any given time \(t\) is identical to its value at the corresponding negative time, \(-t\). This mathematical property directly dictates the visual appearance of an even signal on a graph.
Symmetry of an Even Signal
The condition \(x(t) = x(-t)\) implies that an even signal is symmetrical about the vertical axis (also known as the y-axis). This means if you were to fold the graph of an even signal along the vertical axis, the portion of the graph on the right side of the axis would perfectly align and overlap with the portion on the left side.
- Consider any point \((t, x(t))\) on the graph of an even signal. Due to the symmetry, there will also be a corresponding point \((-t, x(-t))\) where \(x(-t)\) is equal to \(x(t)\). This creates a mirrored image across the vertical axis.
- Common examples of even signals include the cosine function, \(\cos(t)\), because \(\cos(t) = \cos(-t)\). Other examples include polynomial functions with only even powers, such as \(t^2\), \(t^4\), or the absolute value function \(|t|\).
Even Signal Versus Odd Signal
To further clarify the concept of an even signal, it's helpful to briefly compare it with an odd signal:
| Signal Type | Mathematical Condition | Symmetry Property |
|---|---|---|
| Even Signal | \(x(t) = x(-t)\) | Symmetrical about the vertical axis (y-axis) |
| Odd Signal | \(x(t) = -x(-t)\) | Antisymmetric about the origin (rotational symmetry) |
Unlike an even signal, an odd signal exhibits antisymmetry about the origin. This means if you rotate the graph of an odd signal by 180 degrees around the origin, it would perfectly overlap with its original position.
Analyzing the Options for Even Signal
Let's evaluate the given options in the context of an even signal:
- Option 1: symmetrical about the vertical axis. This statement precisely describes the defining characteristic and visual property of an even signal, where \(x(t) = x(-t)\).
- Option 2: antisymmetric about the vertical axis. This is incorrect for an even signal. Antisymmetry about the vertical axis would imply \(x(t) = -x(-t)\) for the vertical axis, which is not the definition of an even signal.
- Option 3: antisymmetric about the horizontal axis. This type of symmetry or antisymmetry is not a standard classification for distinguishing between even and odd signals. Signals are generally not antisymmetric about the horizontal axis in this context.
- Option 4: symmetrical about the horizontal axis. Similar to Option 3, symmetry about the horizontal axis is not a characteristic used to define an even signal in signal processing. This would imply \(x(t) = \pm x(t)\) which is trivial or \(x(t) = \text{constant}\).
Therefore, based on the mathematical definition and visual properties, an even signal is accurately described as symmetrical about the vertical axis.
Important Questions from Standard Signals
Inverse Fourier Transform of δ(ω - ω 0) is ______.
The following statements relate to sampling distributions. Choose the correct code for the statements being correct or incorrect.
Statement I: Sampling distribution of mean is normally distributed irrespective of the type of population distribution and size of samples.
Statement II : The standard deviation of the sampling distribution of mean is less than the standard deviation of the population distribution.
The value of \(\mathop \smallint \limits_{ - \infty }^{ + \infty } {e^{ - t}}\delta \left( {2t - 2} \right)dt\), where \(\delta \left( t \right)\) is the Dirac delta function, is
- \(\mathop \smallint \nolimits_{ - 7}^2 \left( {{t^2} + {t^3} + 1} \right)\delta \left( {t - 3} \right)dt = \_\_\_\_\)
Which of the following points CANNOT be observed about a unit impulse function if it is assumed in the form of a pulse?
