1. Introduction
The Fourier transform assumes that the sampled signal is infinite. This means the final sample point's amplitude should align seamlessly with the point just before the first sample. Ideally, the number of captured sine wave cycles should be an integer. In other words, if the captured signal were repeated sequentially, it would form a continuous waveform. Signals meeting this condition are referred to as coherent.
However, it is not always possible to capture an integer number of sine wave cycles (often a power of two). For the Fourier transform, this implies discontinuity at the signal boundaries, resulting in spectral leakage. This issue can be visualized when switching between coherent and non-coherent signals in a spectral display.
2. Windowing for Non-Coherent Signals
When applying Fourier analysis to non-coherent signals, it becomes difficult to calculate accurate parameters such as SINAD and SNR. Harmonics may also be lost in the frequency spectrum. To mitigate discontinuities, the amplitude of the captured signal can be gradually reduced at the beginning and end. This is the purpose of applying a window function. When a window is applied, it spreads the signal energy across additional frequency bins near the target frequency. The window function is multiplied with the captured signal according to the formula shown below the plot.
Amplitude errors can be observed when comparing results from different window types. For instance, using a rectangular window with a coherent signal, the carrier amplitude matches exactly 1 Vpeak. Switching to a non-coherent signal introduces amplitude errors, as the carrier energy is spread into other frequency bands. This is better visualized in dBc (decibels relative to the carrier). Switching back to Vpeak and selecting a scan window with a coherent signal, the carrier amplitude drops to 0.5 V, with the remaining energy distributed symmetrically in two adjacent bins, maintaining a total of 1 Vpeak. Using a scan window with a non-coherent signal also results in amplitude errors, but they are smaller compared to the rectangular window.
As demonstrated, window functions help reduce spectral leakage and consolidate signal energy into specific adjacent frequency segments. This leads to more accurate parameter calculations and improved visibility of harmonics. The Rife-Vincent 4 window, for example, concentrates nearly all the signal energy into adjacent bins, but this approach is not without trade-offs.
Frequency Resolution Trade-Offs
When displaying the spectrum using dBc with a 0.5 V DC and a coherent signal under the Hamming window, the signal energy is distributed across three bins (the center bin and one on each side). The DC energy occupies the first two bins, while the bin between the DC and the signal represents noise.
Switching to the Rife-Vincent 4 window, the signal energy is distributed across nine bins (excluding DC), centered on the fifth bin with four additional bins on either side. Similarly, the DC occupies the first few bins. Since there are only three bins between the DC and the signal, bins 1 through 4 contain both DC and signal components.
This overlap can also occur between harmonic bins and noise bins. If a harmonic bin's amplitude is less than or equal to its surrounding noise bins, it may absorb or leak energy from those noise bins. As a result, total harmonic distortion (THD) can degrade significantly when the harmonic amplitude is equal to or lower than the noise level while using a window function.
3. Choosing the Right Window Function
One of the key differences between window functions is in their main lobe width and sidelobe characteristics. The main lobe includes the central frequency and its adjacent bins. This can be observed by selecting a coherent signal (with no DC offset) and changing window types. The width of the main lobe affects frequency resolution ¡ª the wider the main lobe, the poorer the resolution but the better the amplitude accuracy.
If amplitude accuracy is the priority, the FlatTop or Blackman-Harris windows are good choices, though their wide main lobes reduce frequency resolution. Hamming and Hanning windows offer better resolution but less accurate amplitude. They have similar shapes and formulas, differing mainly in their sidelobe roll-off rate.
The Rife-Vincent 4 window has extremely low sidelobe levels but a relatively wide main lobe. This makes it suitable for dynamic testing of A/D and D/A converters using single-tone signals. However, due to the broader main lobe, harmonic lobes also become wider, increasing the likelihood of adjacent noise bin leakage into harmonic bins. This can lead to poorer THD measurements in some cases.
Window Characteristics Summary
The following table summarizes key attributes of various window functions. The maximum sidelobe level is the amplitude (in dB) of the highest sidelobe relative to the main lobe. The sidelobe roll-off rate describes how quickly the sidelobe peaks decay with frequency, measured in dB per decade.
