A discrete time real valued signal is said to be symmetric if:

Explanation:

Symmetric Discrete Time Real-Valued Signal

Definition: A discrete-time real-valued signal is said to be symmetric (or even) if it satisfies the condition X(n) = X(-n) for all values of n. This means that the signal remains unchanged when the time index is reversed.

Correct Option Analysis:

The correct option is:

Option 1: X(n) = X(-n)

This option correctly describes the condition for a discrete-time real-valued signal to be symmetric. Symmetry in this context means that the signal's value at time n is the same as its value at time -n. This property is essential in signal processing and analysis, as symmetric signals have specific characteristics that can be exploited in various applications, such as simplifying the computation of Fourier transforms and analyzing the behavior of systems.

Let's analyze the other options to understand why they are incorrect:

Option 2: X(n) = X(n-N)

This option suggests that the signal is periodic with a period of N. While periodicity is an important property in signal processing, it does not imply symmetry. A periodic signal repeats itself after a fixed interval, but it does not necessarily have the property that X(n) = X(-n). Therefore, this option does not describe a symmetric signal.

Option 3: X(n) = -X(n)

This option describes an odd (or antisymmetric) signal. For an odd signal, the value at time n is the negative of its value at time -n. Although odd signals are also important in signal processing, they are not symmetric. Symmetric signals require the condition X(n) = X(-n), not X(n) = -X(n). Therefore, this option is incorrect.

Option 4: X(n) = X(n+N)

This option is another way of describing a periodic signal with a period of N. Similar to Option 2, periodicity does not imply symmetry. A signal that repeats itself after N samples may not have the property that X(n) = X(-n). Therefore, this option does not describe a symmetric signal.

Conclusion:

Understanding the properties of symmetric (even) signals is crucial in the field of signal processing. Symmetric signals have unique characteristics that can be leveraged in various applications, such as simplifying the analysis of systems and signals. The condition X(n) = X(-n) is the defining property of symmetric discrete-time real-valued signals, which distinguishes them from other types of signals such as periodic or odd signals.