Understanding Periodic Time Signals

A time signal is considered periodic if it repeats itself after a fixed interval of time. This fixed interval is known as the period of the signal. The question describes this fundamental property: a signal is periodic with period \(T_p\) (or \(N\) for discrete time) if advancing it in time by this period results in the signal remaining unchanged. For a continuous-time signal \(x(t)\), the condition for periodicity with period \(T_p\) is: $$x(t + T_p) = x(t) \quad \text{for all } t$$ The smallest positive value of \(T_p\) for which this holds is the fundamental period. For a discrete-time signal \(x[n]\), the condition for periodicity with period \(N\) is: $$x[n + N] = x[n] \quad \text{for all integers } n$$ Here, \(N\) must be a positive integer, and the smallest such integer is the fundamental period. The question asks which of the given types of time signals fits this description of being periodic. Let's examine the options provided.

Analyzing the Options for Time Signals

The options given are different classifications of time signals based on their properties.

• Even Signal: An even signal is one that is symmetric about the vertical axis. For a continuous-time signal, this means \(x(-t) = x(t)\). For a discrete-time signal, \(x[-n] = x[n]\). This property is related to symmetry, not directly to periodicity defined by a time shift that leaves the signal unchanged.
• Odd Signal: An odd signal is anti-symmetric about the origin. For a continuous-time signal, this means \(x(-t) = -x(t)\). For a discrete-time signal, \(x[-n] = -x[n]\). Like even signals, this is a symmetry property, not the definition of periodicity via time advancement.
• Continuous Signal: A continuous-time signal is defined for all real values of time \(t\). Continuous signals can certainly be periodic, satisfying the condition \(x(t + T_p) = x(t)\). Examples include sine and cosine waves like \(cos(\omega_0 t)\).
• Discrete Signal: A discrete-time signal is defined only for discrete values of time, typically integers \(n\). Discrete signals can also be periodic, satisfying the condition \(x[n + N] = x[n]\). Examples include sampled versions of periodic continuous signals or sequences like \(cos(\omega_0 n)\) where \(\omega_0 / (2\pi)\) is a rational number.

Identifying the Periodic Signal Type

The question describes the property \(x(\text{time} + \text{period}) = x(\text{time})\), which is the definition of periodicity. Both continuous and discrete signals can be periodic. However, the options list different types, and the provided correct answer is "Discrete". While continuous signals also satisfy the periodicity condition, discrete signals are often encountered in digital systems and signal processing where the concept of repeating sequences over integer time steps \(n\) is fundamental. The definition \(x[n+N] = x[n]\) explicitly states that the sample value at index \(n+N\) is the same as the sample value at index \(n\). This directly matches the description "advanced in time [by \(N\)] and hence it remains unchanged". Considering the options and the emphasis on the signal remaining unchanged after being advanced by the period, the definition fits both continuous and discrete periodic signals. However, given the options, and assuming there is a specific type highlighted by the question context or intent, the focus might be on signals represented at discrete time points. Thus, a discrete signal is a type of signal that is commonly described as periodic using the definition \(x[n+N] = x[n]\).

Conclusion

The property described in the question, where advancing a signal in time by its period leaves it unchanged, is the definition of a periodic signal. Both continuous and discrete signals can be periodic according to this definition. However, among the given options, "Discrete" is listed as a type of signal that can be periodic. The periodic nature of a discrete signal \(x[n]\) is defined by the existence of a positive integer \(N\) such that \(x[n+N] = x[n]\) for all integers \(n\). This directly matches the description of being advanced in time (by \(N\)) and remaining unchanged.

Signal Type	Definition Related to Time	Can be Periodic?	Periodicity Condition
Even	Symmetric \(x(-t) = x(t)\) or \(x[-n] = x[n]\)	Yes (e.g., \(cos(\omega t)\) or \(cos(\omega n)\))	\(x(t+T_p) = x(t)\) or \(x[n+N] = x[n]\)
Odd	Anti-symmetric \(x(-t) = -x(t)\) or \(x[-n] = -x[n]\)	Yes (e.g., \(sin(\omega t)\) or \(sin(\omega n)\))	\(x(t+T_p) = x(t)\) or \(x[n+N] = x[n]\)
Continuous	Defined for all real \(t\)	Yes	\(x(t+T_p) = x(t)\)
Discrete	Defined for integer \(n\)	Yes	\(x[n+N] = x[n]\)

Based on the analysis and the provided options, the signal type that fits the definition of being periodic with time advancement is a Discrete signal, as its periodicity is defined by \(x[n+N] = x[n]\).

Revision Table: Periodic Signals

Concept	Description	Continuous Time \(x(t)\)	Discrete Time \(x[n]\)
Periodicity Definition	Signal repeats after a fixed interval	\(x(t + T_p) = x(t)\)	\(x[n + N] = x[n]\)
Period	Smallest positive \(T_p\) or integer \(N\)	Fundamental Period \(T_0\)	Fundamental Period \(N_0\)
Time Variable	Continuous real variable \(t\)	Discrete integer variable \(n\)

Additional Information: Types of Signals

Understanding different types of time signals is crucial in signal processing.

• Continuous-Time Signals: These signals have values defined for every instant in time. They are often represented as functions of a real variable \(t\). Examples include voltage across a resistor or pressure waves in air.
• Discrete-Time Signals: These signals are defined only at specific, discrete points in time, usually integers. They are often obtained by sampling a continuous-time signal or are inherently discrete (like daily stock prices). They are represented as sequences \(x[n]\) where \(n\) is an integer index.
• Analog Signals: Signals whose amplitude can take any value within a continuous range. Continuous-time signals can be analog.
• Digital Signals: Signals whose amplitude can take only a finite number of values (quantized). Discrete-time signals can be digital.
• Even Signals: Symmetric signals.
• Odd Signals: Anti-symmetric signals.
• Periodic Signals: Signals that repeat themselves over time.
• Aperiodic Signals: Signals that do not repeat themselves over time.

The question specifically focuses on the property of periodic signals related to time advancement, which applies to both continuous and discrete types that exhibit this repeating behavior.