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:
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:
The smallest positive integer \(N\) is called the fundamental period.
Notice that in digital signals the period must always be an integer.
Periodicity of Continuous-Time Sinusoids
Consider the analog sinusoidal signal:
where:
- \(A\) = Amplitude
- \(\omega\) = Angular frequency (rad/s)
- \(\phi\) = Phase angle
For periodicity,
Therefore,
This is the standard formula most students remember.
Example 1
Given:
Here,
Therefore,
Hence, the fundamental period is:
Why This Formula Does Not Directly Work for Digital Signals
Consider the digital signal:
A common mistake is to write:
This is not always correct.
The correct periodicity condition is:
where \(k=1,2,3,\ldots\)
for some integer \(k\). The signal is periodic only if \(N\) becomes an integer.
Periodicity of Discrete-Time Sinusoids
For
the signal is periodic if
is a rational number.
That means:
where \(p\) and \(q\) are integers.
Then the fundamental period is:
after reducing the fraction to its lowest terms.
Example 2
Determine the period of
We have:
Thus,
The fundamental period is:
Example 3
Determine the period of
We have:
Since 3 and 20 are coprime:
Therefore:
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 |
Summary
- \(T=\frac{2\pi}{\omega}\) is valid for continuous-time (analog) sinusoids.
- For discrete-time sinusoids, the correct condition is \(\omega N = 2\pi k\).
- A discrete-time sinusoid is periodic only when \(\frac{\omega}{2\pi}\) is rational.
- Reduce \(\frac{\omega}{2\pi}\) to \(\frac{p}{q}\). The fundamental period is \(q\).
- 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 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):
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.
Note: WSS is assumed because it ensures the autocorrelation depends only on the time difference $\tau$, not the absolute time $t$.