Hartley Transform Calculator

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Overview: Real-Valued Spectra with the Hartley Transform

The Hartley transform is a real-valued alternative to the discrete Fourier transform (DFT). Instead of using complex exponentials with an imaginary unit, it combines sine and cosine into a single real kernel. This makes it especially convenient when your input sequence is purely real and you want a real-valued spectrum.

This page focuses on the discrete Hartley transform (DHT), which is the version used for finite-length sequences. The calculator above takes a real sequence, computes its DHT, and returns the real coefficients that describe the frequency content of the signal.

Definition and Formula

For a real input sequence $x [n]$ of length $N$ , the discrete Hartley transform $H_{k}$ is defined for indices $k = 0, 1, \dots, N - 1$ as:

$H [k] = \sum_{n = 0}^{N - 1} x_{n} cas (\frac{2 π k n}{N})$ where the cas function is $cas (θ) = \cos (θ) + \sin (θ) .$

In expanded form, you can also write the DHT as:

H [k] = \sum_{n = 0}^{N - 1} x [n] (\cos (\frac{2 π k n}{N}) + \sin (\frac{2 π k n}{N}))

A key property of the Hartley transform is that it is its own inverse, up to a scaling factor. The inverse transform has the same mathematical form:

$x [n] = \frac{1}{N} \sum_{k = 0} N^{- 1} H [k] cas (\frac{2 π k n}{N})$

This involutive property (applying the transform twice, with appropriate scaling, gives back the original sequence) makes certain theoretical arguments and some implementations simpler than with the complex DFT.

Relationship to the Discrete Fourier Transform (DFT)

The Hartley transform is closely related to the discrete Fourier transform. If $X_{k}$ is the DFT of the same real sequence $x [n]$ , then the Hartley spectrum $H_{k}$ can be obtained from the real and imaginary parts of $X_{k}$ . A common relationship is:

$H [k] = ℜ {X [k]} - ℑ {X [k]} .$

Conversely, you can reconstruct the DFT from the Hartley spectrum using linear combinations of shifted Hartley coefficients. In practice, this means that any operation you can perform using the DFT (such as convolution, filtering, and spectral analysis) can also be implemented using the DHT, possibly with some algebraic conversion steps.

Because the DHT uses only real arithmetic, it can simplify implementations that work strictly with real-valued signals or hardware that has limited support for complex numbers.

Worked Example

This short example shows exactly how the calculator interprets an input sequence and what the output means.

Input sequence

Suppose you enter the following sequence into the calculator:

1, 0, -1, 0

Here, $N = 4$ and the samples are:

$x [0] = 1$
$x_{1} = 0$
$x [2] = - 1$
$x [3] = 0$

Step-by-step DHT computation

For each output index $k$ , the calculator evaluates

$H [k] = \sum_{n = 0} 3^{} x [n] cas (\frac{2 π k n}{4})$

For $k = 0$ : the angle is always $0$ , and $cas (0) = \cos 0 + \sin 0 = 1$ .

$H [0] = 1 \cdot 1 + 0 \cdot 1 + (- 1) \cdot 1 + 0 \cdot 1 = 0 .$

For $k = 1$ : the angles are $0, \frac{π}{2}, π, \frac{3 π}{2}$ for $n = 0, 1, 2, 3$ respectively. The cas values are:

$cas (0) = 1$
$cas (\frac{π}{2}) = \cos \frac{π}{2} + \sin \frac{π}{2} = 0 + 1 = 1$
$cas (π) = - 1 + 0 = - 1$
$cas (\frac{3 π}{2}) = 0 - 1 = - 1$

Then

$H [1] = 1 \cdot 1 + 0 \cdot 1 + (- 1) \cdot (- 1) + 0 \cdot (- 1) = 2 .$

For $k = 2$ : the angles are $0, π, 2 π, 3 π$ . The cas values are:

$cas (0) = 1$
$cas (π) = - 1 + 0 = - 1$
$cas (2 π) = 1 + 0 = 1$
$cas (3 π) = - 1 + 0 = - 1$

Thus

$H [2] = 1 \cdot 1 + 0 \cdot (- 1) + (- 1) \cdot 1 + 0 \cdot (- 1) = 0 .$

For $k = 3$ : by symmetry for this sequence, you obtain

$H [3] = - 2 .$

The calculator will report the Hartley spectrum as something close to

[0, 2, 0, -2]

(with small rounding differences possible due to floating-point arithmetic).

Interpreting the example

The nonzero coefficients at $k = 1$ and $k = 3$ indicate strong components at those discrete frequency indices, while $k = 0$ and $k = 2$ are effectively zero for this particular pattern. You can experiment with other short sequences in the calculator to see how the spectrum changes when you shift, scale, or combine simple patterns.

How to Use the Hartley Transform Calculator

This calculator is designed to accept real-valued sequences and return the corresponding discrete Hartley transform. Here is how to use it effectively.

Supported input format

Enter numbers separated by commas, for example: 0.5, 1, -0.5, 0.
Optional spaces around commas are ignored: 1, 0, -1, 0 is treated the same as 1,0,-1,0.
Both integers and decimals are allowed: 2, -3.75, 0.001, and so on.
Use a period as the decimal separator (not a comma).
The tool is intended for real values; complex inputs are not supported.

Practical sequence lengths

You can enter short educational examples (length 4, 8, 16) to see structure and symmetry.
Moderate lengths (up to a few thousand points) are usually fine in a modern browser, though computation time increases with $N^{2}$ for a straightforward implementation.
Very long sequences may be slow to compute or display and can be limited by your device and browser.

Reading the output

The calculator outputs an array of Hartley coefficients $H_{k}$ with the same length as the input sequence.
The index $k$ corresponds to a discrete frequency bin. For time-domain sampling interval $T$ and sampling frequency $f_{s} = \frac{1}{T}$ , the frequency associated with index $k$ is $f_k = k f_s / N$ .
The output is real-valued. Positive and negative values indicate the combination of cosine and sine contributions captured by the cas kernel at each frequency.
If you apply the Hartley transform again to the output and divide by $N$ , you recover the original sequence (modulo small numerical errors).

Interpreting Hartley Transform Results

Once you have computed the DHT of a sequence, you can use the coefficients in several ways:

Spectral magnitude intuition: Large absolute values of $H_{k}$ indicate strong oscillatory components at the corresponding discrete frequency index.
Symmetry patterns: For many real-valued sequences with certain symmetries (even or odd), the Hartley spectrum will show predictable patterns and relationships between coefficients.
Filtering in the Hartley domain: Similar to Fourier-domain filtering, you can attenuate or boost specific frequency bins by multiplying $H_{k}$ by a desired frequency response, then applying the inverse transform.

Because the transform is real and its own inverse (apart from scaling), it can be particularly convenient for educational purposes and for implementations where avoiding complex arithmetic is beneficial.

Hartley Transform vs. Fourier Transform

The table below summarizes some practical similarities and differences between the discrete Hartley transform (DHT) and the discrete Fourier transform (DFT) when applied to real input sequences.

Aspect	Discrete Hartley Transform (DHT)	Discrete Fourier Transform (DFT)
Output type	Purely real coefficients $H_{k}$	Complex coefficients $X_{k}$ with real and imaginary parts
Kernel	$c a s (θ) = cos θ + sin θ$	$e^{- j θ} = cos θ - j sin θ$
Inverse relationship	Same form as forward transform with a $\frac{1}{N}$ scale factor	Inverse DFT uses complex conjugate kernel and $\frac{1}{N}$ scale factor
Implementation	Can use only real arithmetic; sometimes advantageous on real-only hardware	Requires complex arithmetic; highly optimized FFT libraries are widely available
Use cases	Educational tools, real-signal processing, systems where complex numbers are inconvenient	General-purpose signal processing, communications, spectral analysis
Conversion between transforms	Can be derived from real and imaginary parts of the DFT	Can be reconstructed from the Hartley spectrum using linear combinations

Assumptions and Limitations of This Calculator

To keep the tool focused and reliable, the implementation is based on several assumptions and has some practical limitations:

Real-valued input only: The calculator assumes all samples are real numbers. Complex inputs are not supported and should be decomposed or converted before use.
No windowing or zero-padding: The transform is applied directly to the sequence you enter. If you need windowing (e.g., Hann, Hamming) or zero-padding to a specific length, apply those steps to your data before pasting it into the tool.
Standard normalization: The forward transform is unscaled, and the inverse transform (if implemented separately) would include a $\frac{1}{N}$ factor. Be aware of this convention when comparing to other software that might distribute scaling differently.
Numerical precision: Results are computed using standard double-precision floating-point arithmetic. Very large magnitudes, extremely small differences, or long sequences may accumulate rounding errors.
Computation time: For long sequences, the direct DHT computation can be slow because it scales approximately with $N^{2}$ . If performance is critical, consider using specialized libraries or fast Hartley transform (FHT) algorithms.
Interpretation responsibility: The calculator returns numerical coefficients but does not automatically classify peaks, detect patterns, or design filters. You are responsible for interpreting the spectrum in the context of your own application.

Understanding these assumptions will help you interpret the output correctly and decide when this tool is appropriate for your task.

Applications and Further Exploration

The Hartley transform can be applied to many of the same problems as the Fourier transform:

Signal analysis: Inspecting the frequency content of sampled audio, vibration, or sensor data without dealing directly with complex numbers.
Image processing: Applying real-valued transforms to grayscale images or intensity profiles, especially in educational contexts.
Filtering and convolution: Implementing filters via multiplication in the Hartley domain and then transforming back to the time domain.
Teaching and visualization: Demonstrating the connection between sine/cosine components and spectrum without introducing complex arithmetic at an early stage.

You can experiment with simple sequences in the calculator to see how different operations (like shifting, scaling, or combining signals) affect the Hartley spectrum and to build intuition for real-valued spectral analysis.

Hartley Transform Calculator

Overview: Real-Valued Spectra with the Hartley Transform

Definition and Formula

Relationship to the Discrete Fourier Transform (DFT)

Worked Example

Input sequence

Step-by-step DHT computation

Interpreting the example

How to Use the Hartley Transform Calculator

Supported input format

Practical sequence lengths

Reading the output

Interpreting Hartley Transform Results

Hartley Transform vs. Fourier Transform

Assumptions and Limitations of This Calculator

Applications and Further Exploration

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