Rectangular Signal

Introduction

The rectangular pulse (also called the gate function or rect function) is one of the most important signals in both theory and practice. It represents a signal that is "on" for a finite duration and "off" otherwise — a model for digital bits, radar pulses, sampling windows, and countless other situations where something happens for a limited time. What makes the rectangular pulse especially important in signals and systems is its beautiful Fourier transform relationship: a rectangle in time transforms to a sinc function in frequency, and this duality appears throughout signal processing.

For B.Tech students, understanding the rectangular pulse thoroughly prepares you for sampling theory, windowing in spectral analysis, communication systems (pulse shaping), and filter design. It's a signal you'll encounter in virtually every chapter of this course.

Mathematical Definition

Unit Rectangular Pulse

The unit rectangular pulse centered at the origin with width $T$ is defined as:

$$\text{rect}\left(\frac{t}{T}\right) = \begin{cases} 1 & |t| < T/2 \\ 1/2 & |t| = T/2 \\ 0 & |t| > T/2 \end{cases}$$

The simpler unit-width version is:

$$\text{rect}(t) = \begin{cases} 1 & |t| < 1/2 \\ 0 & |t| > 1/2 \end{cases}$$

It can also be expressed using unit step functions:

$$\text{rect}\left(\frac{t}{T}\right) = u(t + T/2) - u(t - T/2)$$

This representation is very useful for analysis — it shows the rectangular pulse as the difference of two shifted step functions.

Properties of the Rectangular Pulse

Amplitude: The peak value is 1 (for the unit rect).

Duration: The pulse has width $T$ (from $-T/2$ to $T/2$).

Area: $\int_{-\infty}^{\infty} \text{rect}(t/T) \, dt = T$

Energy: $E = \int_{-\infty}^{\infty} |\text{rect}(t/T)|^2 dt = T$

Even symmetry: $\text{rect}(t/T)$ is an even function — symmetric about $t = 0$.

The Sinc Function: Fourier Transform of Rect

The Fourier transform of the rectangular pulse is the celebrated sinc function:

$$\mathcal{F}\left\{\text{rect}\left(\frac{t}{T}\right)\right\} = T \cdot \text{sinc}(fT)$$

where the normalized sinc function is defined as:

$$\text{sinc}(x) = \frac{\sin(\pi x)}{\pi x}$$

(Note: some textbooks use the unnormalized version $\text{sinc}(x) = \sin(x)/x$.)

Derivation

Starting from the Fourier transform definition:

$$X(f) = \int_{-\infty}^{\infty} \text{rect}(t/T) \cdot e^{-j2\pi ft} dt = \int_{-T/2}^{T/2} e^{-j2\pi ft} dt$$

$$= \left[\frac{e^{-j2\pi ft}}{-j2\pi f}\right]_{-T/2}^{T/2} = \frac{e^{-j\pi fT} - e^{j\pi fT}}{-j2\pi f}$$

$$= \frac{2\sin(\pi fT)}{2\pi f} = T \cdot \frac{\sin(\pi fT)}{\pi fT} = T \cdot \text{sinc}(fT)$$

Properties of the Sinc Function

The sinc function has these important characteristics:

• Peak value: $\text{sinc}(0) = 1$ (by L'Hôpital's rule)
• Zero crossings: At $x = \pm 1, \pm 2, \pm 3, \ldots$ (frequencies $f = \pm 1/T, \pm 2/T, \ldots$)
• Main lobe width: $2/T$ in frequency — wider pulses in time give narrower main lobes (time-frequency reciprocity)
• Sidelobes: Decay as $1/(\pi x)$ — relatively slow decay, which is why windowing is needed in practice
• Area: $\int_{-\infty}^{\infty} \text{sinc}(x) dx = 1$

Time-Frequency Reciprocity

This is a crucial insight: a rectangular pulse of width $T$ has a sinc spectrum with main lobe width $2/T$:

• Narrow pulse (small $T$) → Wide spectrum (large bandwidth)
• Wide pulse (large $T$) → Narrow spectrum (small bandwidth)

This is the uncertainty principle of signal processing — you cannot simultaneously have a signal that is compact in both time and frequency. The rectangular pulse makes this trade-off explicit and quantifiable.

Bandwidth of a Rectangular Pulse

The first-null bandwidth is defined as the frequency of the first zero crossing:

$$B_{null} = \frac{1}{T}$$

The 3-dB bandwidth (where the sinc magnitude drops to $1/\sqrt{2}$) is approximately:

$$B_{3dB} \approx \frac{0.886}{T}$$

Duality: Sinc in Time, Rect in Frequency

By the duality property of the Fourier transform, if rect transforms to sinc, then sinc in time transforms to rect in frequency:

$$\mathcal{F}\{T \cdot \text{sinc}(Bt)\} = \frac{1}{B}\text{rect}\left(\frac{f}{B}\right)$$

This is the ideal low-pass filter! A signal with a sinc impulse response has a perfectly rectangular (brick-wall) frequency response — passing all frequencies below $B/2$ and blocking all frequencies above. The practical challenge is that the sinc function extends infinitely in both time directions, making ideal filters non-causal and unrealizable.

Convolution of Two Rectangles: The Triangle

The convolution of a rectangular pulse with itself produces a triangular pulse:

$$\text{rect}(t/T) * \text{rect}(t/T) = T \cdot \text{tri}(t/T)$$

where $\text{tri}(t/T)$ is the triangle function with width $2T$. In the frequency domain, this corresponds to:

$$\text{sinc}^2(fT)$$

The triangular pulse has faster spectral decay ($1/f^2$ vs $1/f$) because it is smoother (continuous first derivative vs. discontinuous).

Discrete-Time Rectangular Sequence

The discrete-time rectangular window of length $N$ is:

$$w[n] = \begin{cases} 1 & 0 \leq n \leq N-1 \\ 0 & \text{otherwise} \end{cases}$$

Its DTFT is:

$$W(e^{j\omega}) = e^{-j\omega(N-1)/2} \cdot \frac{\sin(N\omega/2)}{\sin(\omega/2)}$$

The function $\sin(N\omega/2)/\sin(\omega/2)$ is called the Dirichlet kernel or digital sinc. It has a main lobe width of $4\pi/N$ and sidelobes approximately 13 dB below the main lobe — this is why the rectangular window has poor spectral leakage characteristics and other windows (Hamming, Hanning, Blackman) are preferred in spectral analysis.

Worked Examples

Example 1: Find the Fourier transform of $x(t) = 5\text{rect}((t-2)/4)$.

Solution: This is a rectangular pulse with amplitude 5, width 4, centered at $t = 2$.

Using the time-shifting property: $$X(f) = 5 \cdot 4 \cdot \text{sinc}(4f) \cdot e^{-j2\pi f \cdot 2} = 20\text{sinc}(4f)e^{-j4\pi f}$$

Example 2: Find the bandwidth of a 1 µs radar pulse.

Solution: For $T = 1 \times 10^{-6}$ s: $$B_{null} = 1/T = 1 \text{ MHz}$$

The main lobe extends from $-1$ MHz to $+1$ MHz, so the total null-to-null bandwidth is 2 MHz.

Example 3: Determine the energy of $x(t) = 3\text{rect}(t/2)$.

Solution: $E = \int_{-1}^{1} |3|^2 dt = 9 \times 2 = 18$ joules.

Alternatively, using Parseval's theorem: $E = \int_{-\infty}^{\infty} |X(f)|^2 df = \int_{-\infty}^{\infty} |6\text{sinc}(2f)|^2 df = 36 \cdot \frac{1}{2} = 18$ ✓

Applications

• Digital communications: Rectangular pulses represent binary data; their sinc spectrum determines bandwidth requirements
• Radar: Transmitted pulse shape determines range resolution ($\Delta R = cT/2$)
• Sampling: The sampling process can be modeled using a train of narrow rectangular pulses
• Windowing: The rectangular window in spectral analysis (simplest but worst sidelobe behavior)
• Signal gating: Extracting a portion of a signal by multiplying with a rect function

Key Takeaways

• The rectangular pulse $\text{rect}(t/T)$ is 1 for $|t| < T/2$ and 0 otherwise
• Its Fourier transform is $T\text{sinc}(fT)$ — the sinc function with zeros at multiples of $1/T$
• Time-frequency reciprocity: narrow pulse → wide bandwidth, and vice versa
• The duality $\text{sinc}(t) \leftrightarrow \text{rect}(f)$ gives the ideal (unrealizable) low-pass filter
• Rect convolved with rect gives triangle; sinc squared in frequency
• The discrete rectangular window has 13 dB sidelobe level — limiting spectral analysis resolution