Triangular Signal

Introduction

The triangular signal (also called the triangle function or tri function) is a piecewise-linear signal that rises linearly to a peak and then falls linearly back to zero. It appears throughout signal processing as the result of convolving a rectangular pulse with itself, as the autocorrelation function of a rectangular pulse, as a window function in spectral analysis, and as the basis for linear interpolation in signal reconstruction. Understanding the triangular signal connects convolution, correlation, spectral analysis, and practical applications in a single elegant framework.

For B.Tech students, the triangular signal demonstrates how convolution smooths discontinuities (rect has sharp corners, triangle does not), and its $\text{sinc}^2$ Fourier transform shows improved spectral decay compared to the rectangular pulse.

Mathematical Definition

Unit Triangle Function

The unit triangle function (also called the unit triangular pulse) centered at the origin is defined as:

$$\text{tri}(t) = \Lambda(t) = \begin{cases} 1 - |t| & |t| \leq 1 \\ 0 & |t| > 1 \end{cases}$$

This has peak value 1 at $t = 0$, linearly decreases to 0 at $t = \pm 1$, giving a total base width of 2.

A general triangle with width $2T$ and height 1:

$$\text{tri}\left(\frac{t}{T}\right) = \begin{cases} 1 - |t|/T & |t| \leq T \\ 0 & |t| > T \end{cases}$$

Expression Using Ramp Functions

The triangle can be expressed using ramp functions:

$$\text{tri}(t) = r(t+1) - 2r(t) + r(t-1)$$

This shows the triangle as a second difference of the ramp — each ramp adds a slope of 1, and the combination creates the rising and falling linear segments.

Alternatively: $\text{tri}(t) = (t+1)u(t+1) - 2t \cdot u(t) + (t-1)u(t-1)$

Properties

Even Symmetry

The triangle function is an even function: $\text{tri}(-t) = \text{tri}(t)$. Its Fourier transform is therefore purely real.

Area

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

(The area of a triangle with base $2T$ and height 1 is $\frac{1}{2} \times 2T \times 1 = T$.)

Energy

$$E = \int_{-\infty}^{\infty} |\text{tri}(t/T)|^2 \, dt = \int_{-T}^{T}(1-|t|/T)^2 dt = \frac{2T}{3}$$

Continuity

Unlike the rectangular pulse which has discontinuities at its edges, the triangular pulse is continuous everywhere. Its first derivative, however, has discontinuities (at $t = -T, 0, T$). This additional smoothness compared to rect results in faster spectral decay.

Fourier Transform

The Fourier transform of the unit triangle function is:

$$\mathcal{F}\{\text{tri}(t/T)\} = T \cdot \text{sinc}^2(fT)$$

Derivation via Convolution

Since $\text{tri}(t/T) = \frac{1}{T}\text{rect}(t/T) * \text{rect}(t/T)$, and convolution in time corresponds to multiplication in frequency:

$$\mathcal{F}\{\text{tri}(t/T)\} = \frac{1}{T} \cdot [T\text{sinc}(fT)]^2 = T\text{sinc}^2(fT)$$

Spectral Properties

The $\text{sinc}^2$ spectrum has important advantages over the sinc spectrum of the rectangle:

• All positive: $\text{sinc}^2(x) \geq 0$ for all $x$ (no negative sidelobes)
• Faster decay: Main envelope decays as $1/f^2$ instead of $1/f$
• Same zero crossings: At $f = \pm n/T$ for integer $n \neq 0$
• Narrower main lobe: Same width as sinc ($2/T$), but better concentrated

Relationship to Convolution

Rect * Rect = Triangle

This is one of the most important convolution results:

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

Convolution of two identical rectangles produces a triangle. This can be verified graphically by the "flip and slide" method:

• For $-T < t < 0$: overlap increases linearly from 0 to $T$
• For $0 < t < T$: overlap decreases linearly from $T$ to 0
• For $|t| > T$: no overlap, result is 0

The triangle's peak value is $T$ (the area of the rect), occurring when the two rectangles perfectly overlap.

Autocorrelation of Rectangular Pulse

The autocorrelation of a signal $x(t)$ is $R_{xx}(\tau) = x(\tau) * x^*(-\tau)$. For a real rect function:

$$R_{xx}(\tau) = \text{rect}(t/T) * \text{rect}(-t/T) = \text{rect}(t/T) * \text{rect}(t/T) = T \cdot \text{tri}(\tau/T)$$

(Since rect is even, $\text{rect}(-t) = \text{rect}(t)$.) This demonstrates that the autocorrelation of a rectangular pulse is triangular — a result frequently used in radar signal processing and correlation receivers.

The Bartlett (Triangular) Window

In spectral analysis, the triangular window (Bartlett window) is:

$$w[n] = 1 - \frac{|n - (N-1)/2|}{(N-1)/2}, \quad 0 \leq n \leq N-1$$

Compared to the rectangular window:

• Sidelobe level: -26 dB (vs -13 dB for rectangular) — better spectral leakage suppression
• Main lobe width: $4\pi/N$ (vs $2\pi/N$) — twice as wide, reducing frequency resolution
• No abrupt transitions: Smooth tapering reduces spectral leakage

The Bartlett window is equivalent to convolving the rectangular window's spectrum with itself (in frequency), which is why sidelobes are reduced (squaring makes everything positive and narrows the main lobe in relative terms).

Triangular Wave (Periodic)

A periodic triangular wave with period $T_0$ and amplitude $A$ has the Fourier series:

$$x(t) = \frac{8A}{\pi^2}\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2}\cos((2k+1)\omega_0 t)$$

Key observations:

• Only odd harmonics are present (like the square wave)
• Coefficients decay as $1/k^2$ (vs $1/k$ for square wave) — faster convergence because the triangle is smoother (continuous, with no discontinuities)
• The factor $(-1)^k$ alternates the sign of successive harmonics

Worked Examples

Example 1: Find the Fourier transform of $x(t) = 4\text{tri}(t/3)$.

Solution: $X(f) = 4 \times 3 \times \text{sinc}^2(3f) = 12\text{sinc}^2(3f)$

First null at $f = 1/3$ Hz. Peak value $X(0) = 12$.

Example 2: A rectangular pulse of width 2 ms is convolved with itself. What is the result?

Solution: $\text{rect}(t/T) * \text{rect}(t/T) = T \cdot \text{tri}(t/T)$ with $T = 2$ ms.

Result is a triangular pulse with base width 4 ms, peak value 2 ms (at $t = 0$).

Example 3: Verify the energy of $x(t) = \text{tri}(t)$ using Parseval's theorem.

Time domain: $E = \int_{-1}^{1}(1-|t|)^2 dt = 2\int_0^1(1-t)^2 dt = 2[-(1-t)^3/3]_0^1 = 2/3$

Frequency domain: $E = \int_{-\infty}^{\infty}|\text{sinc}^2(f)|^2 df = \int_{-\infty}^{\infty}\text{sinc}^4(f)df = 2/3$ ✓

Applications

• Linear interpolation: Reconstructing a signal from samples using triangle-weighted averaging
• Bartlett window: Spectral analysis with reduced leakage (vs rectangular window)
• Matched filter output: The output of a matched filter for a rectangular pulse is triangular
• PWM generation: Comparing a signal with a triangular carrier generates pulse-width modulated waveforms
• Sampling theory: The zero-order hold output convolved with itself gives a first-order hold (triangular interpolation)
• Correlation receivers: Radar and communication systems use triangular correlation peaks for timing estimation

Key Takeaways

• The triangle function $\text{tri}(t/T)$ has peak 1, base width $2T$, and is continuous everywhere
• It is the convolution of a rect with itself: $\text{rect} * \text{rect} = T \cdot \text{tri}$
• Its Fourier transform is $T\text{sinc}^2(fT)$ — non-negative spectrum with $1/f^2$ decay
• Smoother than rect → faster spectral decay → better sidelobe behavior
• Used as the Bartlett window in spectral analysis (26 dB sidelobe suppression)
• The autocorrelation of a rectangular pulse is triangular — fundamental in radar/communications
• The periodic triangular wave has Fourier coefficients decaying as $1/k^2$ (only odd harmonics)