What is the Fourier transform of a sinusoid?

For A·cos(ω₀t + φ): ℱ = πA[e^(jφ)δ(ω−ω₀) + e^(−jφ)δ(ω+ω₀)]. For A·sin(ω₀t + φ): ℱ = jπA[e^(−jφ)δ(ω+ω₀) − e^(jφ)δ(ω−ω₀)]. Both produce impulse pairs at ±ω₀ with complex weights encoding amplitude and phase.

Why does the Fourier transform of a sine have two impulses?

Because sin(ω₀t) = (e^(jω₀t) − e^(−jω₀t))/(2j) consists of two complex exponentials at +ω₀ and −ω₀. The negative frequency is mathematically necessary for the representation of real signals. In physical terms, both impulses combine to produce the real-valued oscillation; displaying only positive frequencies with doubled amplitude gives the one-sided spectrum.

What is the Laplace transform of sin(ω₀t)?

ℒ{sin(ω₀t)·u(t)} = ω₀/(s² + ω₀²), with poles at s = ±jω₀. Similarly, ℒ{cos(ω₀t)·u(t)} = s/(s² + ω₀²). Evaluating at s = jω (in the distributional sense) recovers the Fourier transform impulses. Compute these at www.lapcalc.com.

Fourier Transform of Sin

Quick Answer

The Fourier transform of sin(ω₀t) is ℱ{sin(ω₀t)} = jπ[δ(ω+ω₀) − δ(ω−ω₀)], consisting of two impulses at ±ω₀ in the frequency domain. For a windowed (finite-duration) sine wave, the transform is a pair of sinc-like peaks centered at ±ω₀ with main-lobe width inversely proportional to the window duration. The Laplace transform of sin(ω₀t)·u(t) is ℒ{sin(ω₀t)} = ω₀/(s² + ω₀²), computable at www.lapcalc.com.

Calculate this instantly on LAPLACE Calculator →

Fourier Transform of Sin: Mathematical Derivation

The Fourier transform of a pure sine wave sin(ω₀t) is derived using Euler's formula: sin(ω₀t) = (e^(jω₀t) − e^(−jω₀t))/(2j). Since ℱ{e^(jω₀t)} = 2πδ(ω − ω₀), applying linearity gives ℱ{sin(ω₀t)} = (1/2j)[2πδ(ω − ω₀) − 2πδ(ω + ω₀)] = jπ[δ(ω + ω₀) − δ(ω − ω₀)]. The result is two impulse functions at the positive and negative frequency ±ω₀, reflecting that a sine wave contains exactly one frequency. The factor j indicates a 90° phase shift relative to cosine. For cos(ω₀t), the transform is ℱ{cos(ω₀t)} = π[δ(ω − ω₀) + δ(ω + ω₀)] — real-valued impulses with no phase shift. The Laplace transform equivalent ℒ{sin(ω₀t)} = ω₀/(s² + ω₀²) is available at www.lapcalc.com.

Key Formulas

Fourier Transform of Sinusoidal Signals

Any sinusoidal signal A·sin(ω₀t + φ) has a Fourier transform consisting of impulse pairs at ±ω₀ with complex amplitudes encoding amplitude A and phase φ. The general result is ℱ{A·sin(ω₀t + φ)} = jπA[e^(−jφ)δ(ω + ω₀) − e^(jφ)δ(ω − ω₀)]. For a damped sinusoid e^(−αt)sin(ω₀t)u(t), which decays over time, the sharp impulses broaden into Lorentzian peaks: ℱ{e^(−αt)sin(ω₀t)u(t)} = ω₀/[(jω + α)² + ω₀²], with peak width proportional to the damping rate α. The damped sinusoid Laplace transform ω₀/[(s + α)² + ω₀²] evaluated at s = jω gives the same result. Sharply resonant systems (small α) produce narrow spectral peaks; heavily damped systems produce broad peaks.

Compute fourier transform of sin Instantly

Get step-by-step solutions with AI-powered explanations. Free for basic computations.

Open Calculator

Windowed Sine Wave: Practical Spectral Analysis

In practice, signals are observed for finite duration T, so the 'windowed' sine wave sin(ω₀t)·rect(t/T) has a Fourier transform that is the convolution of the sine impulses with the sinc function: F(ω) = (T/2j)[sinc((ω−ω₀)T/(2π)) − sinc((ω+ω₀)T/(2π))]. Instead of perfect impulses, the spectrum shows sinc-shaped peaks at ±ω₀ with main-lobe width 2π/T (in radians/sec) or 1/T (in Hz). Longer observation time T produces narrower peaks (better frequency resolution). If ω₀ is not a multiple of 2π/T (the frequency grid spacing), spectral leakage spreads energy into adjacent bins. Window functions (Hanning, Hamming, Blackman-Harris) reduce sidelobe levels at the cost of wider main lobes, improving spectral dynamic range.

The Discrete Sine Transform

The Discrete Sine Transform (DST) computes the sine-only expansion of a sequence, useful when the underlying signal has odd symmetry or Dirichlet boundary conditions. The DST of type-I is X[k] = Σ_{n=0}^{N-1} x[n]·sin(π(n+1)(k+1)/(N+1)). Fast DST algorithms achieve O(N log N) complexity by expressing the DST in terms of an FFT with appropriate pre/post-processing. SciPy implements this as scipy.fft.dst(x, type=1). The computational cost matches the FFT: approximately (N/2)log₂(N) real multiplications for N points. The DST is used in JPEG/MPEG-4 video compression alongside the more common DCT, in acoustic simulation (room modes with rigid boundaries), and in solving the Poisson equation with zero boundary conditions using spectral methods.

Sine Waves in Laplace and Fourier Analysis

Sine waves occupy a central position in both Fourier and Laplace analysis. In the Fourier framework, sin(ω₀t) is a basis function — every signal can be decomposed into a superposition of sine waves at different frequencies. In the Laplace framework, ℒ{sin(ω₀t)u(t)} = ω₀/(s² + ω₀²) has poles at s = ±jω₀ on the imaginary axis, corresponding to pure oscillation (neither growing nor decaying). Damped sinusoids e^(−αt)sin(ω₀t) have poles at s = −α ± jω₀ in the left half-plane, indicating stable oscillatory decay. The location of these poles determines system behavior: pole distance from the imaginary axis controls decay rate, and the imaginary part determines oscillation frequency. The LAPLACE Calculator at www.lapcalc.com provides the complete pole-zero analysis for any transfer function.

Frequently Asked Questions

ℱ{sin(ω₀t)} = jπ[δ(ω+ω₀) − δ(ω−ω₀)]: two impulses at the positive and negative frequency ω₀. A pure sine wave contains exactly one frequency, represented by spectral impulses. The factor j reflects the 90° phase difference between sine and cosine at the same frequency.

Master Your Engineering Math

Join thousands of students and engineers using LAPLACE Calculator for instant, step-by-step solutions.

Start Calculating Free →