Hartley Transform Calculator - Real Fourier Alternative

Real-Valued Spectrum via the Hartley Transform

The Hartley transform is a lesser-known cousin of the discrete Fourier transform (DFT) that maps a real sequence to a real spectrum. Whereas the DFT produces complex values involving $i$ , the Hartley transform replaces complex exponentials with a combination of sine and cosine. For a sequence $x_{n}$ of length $N$ , the discrete Hartley transform (DHT) is defined by

$H_{k} = \sum_{n}^{0 N - 1} x_{n} (cos (\frac{2 π k n}{N}) + sin (\frac{2 π k n}{N}))$

This expression uses the "cas" function, $cas (θ) = \cos (θ) + \sin (θ)$ . Because it avoids complex arithmetic, the DHT can be faster when the signal is purely real. Additionally, it is its own inverse up to a scaling factor, simplifying transform pairs.

Computing the Transform

To compute the DHT, we apply the above sum for each index $k$ . The inverse transform has the same form divided by $N$ , making the Hartley transform an involution. Our calculator parses the sequence you enter, evaluates the cas function at each product of indices, and outputs the resulting array. Internally we rely on the math.js library for trigonometric functions.

Comparison to the Fourier Transform

While the DFT expresses a signal as a sum of complex exponentials, the DHT uses only real trigonometric components. The two transforms are closely related: each can be derived from the other through simple linear combinations. Specifically, if $X_{k}$ denotes the DFT, then $H_{k}$ equals $Re (X_{k}) - Im (X_{k})$ . Conversely, one can recover the DFT by averaging two shifted Hartley transforms. This equivalence means that any application of the DFT—spectral analysis, convolution, filtering—can also be performed using the DHT.

Historical Context

The transform was introduced by Ralph Hartley in the 1940s and gained renewed interest in the 1980s when fast Hartley transform (FHT) algorithms were developed. By avoiding complex numbers, these algorithms reduced memory usage and simplified hardware implementations. Though modern processors easily handle complex arithmetic, the Hartley transform remains a valuable teaching tool for understanding the interplay between sine and cosine components.

Signal Processing Applications

Because the Hartley transform is its own inverse (aside from scaling), filtering operations become symmetric. Convolving two real sequences corresponds to element-wise multiplication of their Hartley spectra, much like with the Fourier transform. Audio engineers and image processors can thus apply real-valued convolutions using the DHT, obtaining the same results as with the DFT but with potentially reduced computational overhead.

Using the Calculator

Enter a sequence of numbers separated by commas, such as 1,0,-1,0. The calculator computes the transform for each index and lists the results. Try simple patterns to see how symmetry in the input manifests in the output. Because the cas function combines sine and cosine, even sequences produce purely real results that mirror the frequencies present.

Understanding the Output

The Hartley transform coefficients indicate the strength of each frequency component relative to the cas basis functions. Peaks correspond to strong correlations between the sequence and those sinusoids. By inspecting the spectrum, you can infer periodicity, filter noise, or compress data. The transform's real nature provides immediate intuition without complex-phase interpretation.

Further Exploration

To deepen your understanding, experiment with random sequences, smooth windows, or known test signals. Try reconstructing the original data by applying the inverse transform, scaling by $1 / N$ . Observe how truncating or modifying spectral coefficients affects the reconstructed sequence. Such exercises highlight the power of frequency-domain analysis.

Computational Complexity

Like the Fast Fourier Transform, the fast Hartley transform reduces the naive (O(N^2)) cost of evaluating the sums to approximately (O(N log N)). Modern libraries exploit symmetries so that the additional sine terms do not double the work. On vector processors and real-time systems where multiplication and addition are expensive, the cas formulation can offer tangible speed benefits. Our calculator performs a straightforward (O(N^2)) computation for clarity, but you can explore optimized implementations in dedicated signal-processing packages.

Real-World Example

Imagine analyzing vibration data from a rotating machine. If the sampled sequence has strong periodic components, the Hartley spectrum will reveal spikes at frequencies corresponding to mechanical resonances. By examining these peaks you can diagnose imbalance or wear. Because only real arithmetic is required, embedded sensors can perform this analysis using modest hardware. The Hartley transform thus provides practical insight in scenarios where simplicity and speed matter.