Periodicity of Continuous and Discrete Signals

Understanding Periodicity of Sinusoidal Signals: Continuous-Time vs Discrete-Time Signals

Periodicity is one of the most important concepts in Signals and Systems, and Digital Signal Processing (DSP). Many students know the formula \(T=\frac{2\pi}{\omega}\), but often get confused when similar questions appear in the context of discrete-time signals.

In this article, we will clearly understand the difference between periodicity in continuous-time (analog) signals and discrete-time (digital) signals with examples and exam-oriented shortcuts.

What is a Periodic Signal?

A signal is said to be periodic if it repeats itself after a fixed interval.

Continuous-Time (Analog) Signal

A continuous-time signal \(x(t)\) is periodic if there exists a positive number \(T\) such that:

\[ x(t+T)=x(t) \]

The smallest positive value of \(T\) is called the fundamental period.

Discrete-Time (Digital) Signal

A discrete-time signal \(x[n]\) is periodic if there exists a positive integer \(N\) such that:

\[ x[n+N]=x[n] \]

The smallest positive integer \(N\) is called the fundamental period.

Notice that in digital signals the period must always be an integer.

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Periodicity of Continuous-Time Sinusoids

Consider the analog sinusoidal signal:

\[ x(t)=A\cos(\omega t+\phi) \]

where:

• \(A\) = Amplitude
• \(\omega\) = Angular frequency (rad/s)
• \(\phi\) = Phase angle

For periodicity,

\[ \omega T = 2\pi \]

Therefore,

\[ T=\frac{2\pi}{\omega} \]

This is the standard formula most students remember.

Example 1

Given:

\[ x(t)=5\cos(20t) \]

Here,

\[ \omega=20\ \text{rad/s} \]

Therefore,

\[ T=\frac{2\pi}{20} =\frac{\pi}{10} \]

Hence, the fundamental period is:

\[ \boxed{\frac{\pi}{10}\ \text{seconds}} \]

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Why This Formula Does Not Directly Work for Digital Signals

Consider the digital signal:

\[ x[n]=A\cos(\omega n+\phi) \]

A common mistake is to write:

\[ N=\frac{2\pi}{\omega} \]

This is not always correct.

The correct periodicity condition is:

\[ \omega N = 2\pi k \]

where \(k=1,2,3,\ldots\)

\[ N=\frac{2\pi k}{\omega} \]

for some integer \(k\). The signal is periodic only if \(N\) becomes an integer.

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Periodicity of Discrete-Time Sinusoids

For

\[ x[n]=A\cos(\omega n+\phi) \]

the signal is periodic if

\[ \frac{\omega}{2\pi} \]

is a rational number.

That means:

\[ \frac{\omega}{2\pi} = \frac{p}{q} \]

where \(p\) and \(q\) are integers.

Then the fundamental period is:

\[ N=q \]

after reducing the fraction to its lowest terms.

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Example 2

Determine the period of

\[ x[n]=\cos\left(\frac{\pi}{4}n\right) \]

We have:

\[ \frac{\omega}{2\pi} = \frac{\pi/4}{2\pi} = \frac{1}{8} \]

Thus,

\[ N=8 \]

The fundamental period is:

\[ \boxed{8} \]

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Example 3

Determine the period of

\[ x[n]=\cos\left(\frac{3\pi}{10}n\right) \]

We have:

\[ \frac{\omega}{2\pi} = \frac{3\pi/10}{2\pi} = \frac{3}{20} \]

Since 3 and 20 are coprime:

\[ N=20 \]

Therefore:

\[ \boxed{\text{Fundamental Period}=20} \]

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Continuous-Time vs Discrete-Time Periodicity

Property	Continuous-Time Signal	Discrete-Time Signal
Signal Form	\(A\cos(\omega t+\phi)\)	\(A\cos(\omega n+\phi)\)
Period Condition	\(\omega T=2\pi\)	\(\omega N=2\pi k\)
Period Formula	\(T=\frac{2\pi}{\omega}\)	\(N=\frac{2\pi k}{\omega}\)
Period Value	Real Number	Integer
Requirement	Always Periodic	\(\omega/2\pi\) must be rational

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Summary

1. \(T=\frac{2\pi}{\omega}\) is valid for continuous-time (analog) sinusoids.
2. For discrete-time sinusoids, the correct condition is \(\omega N = 2\pi k\).
3. A discrete-time sinusoid is periodic only when \(\frac{\omega}{2\pi}\) is rational.
4. Reduce \(\frac{\omega}{2\pi}\) to \(\frac{p}{q}\). The fundamental period is \(q\).
5. For the GATE-style example, both \(\boxed{0.1\pi}\) and \(\boxed{0.3\pi}\) produce a fundamental period of 20.

Autocorrelation and Periodicity

The Autocorrelation function measures the correlation of a signal with a delayed version of itself. It is a powerful tool for finding repeating patterns or periodicity in a noisy signal.

For a random process: $R_{xx}(\tau) = E[x(t)x(t+\tau)]$
For an energy signal: $R_{xx}(\tau) = \int_{-\infty}^{\infty} x(t)x(t+\tau) dt$

When a signal $x(t)$ is transmitted through a wireless medium, it often picks up Additive White Gaussian Noise (AWGN):

$y(t) = x(t) + n(t)$

Crucially, the periodicity of the autocorrelation of $y(t)$ remains the same as $x(t)$. Autocorrelation effectively "filters" the random noise components, which only contribute to the signal at zero lag ($\tau = 0$).

Wiener–Khinchin Theorem

This theorem links the time domain to the frequency domain by stating that the Power Spectral Density (PSD) of a Wide-Sense Stationary (WSS) process is the Fourier Transform of its autocorrelation function.

$S_X(f) = \mathcal{F}\{R_X(\tau)\} = \int_{-\infty}^{\infty} R_X(\tau) e^{-j2\pi f \tau} d\tau$

Note: WSS is assumed because it ensures the autocorrelation depends only on the time difference $\tau$, not the absolute time $t$.

Read More: about Autocorrelation and Periodicity

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