Keywords: coherent sampling, window sampling, high-speed, data converters, analog-to-digital, analogue-to-digital, A to D, A/D, converters, ADCs
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TUTORIALS 1040

Abstract: One of the most useful techniques for evaluating the dynamic performance of fast and ultra-fast data converters is coherent sampling. This technique increases the spectral resolution of a Fast Fourier Transform (FFT) and eliminates the need for window sampling when certain conditions are met. However, if the conditions for coherent sampling cannot be met, window sampling can be used. The following application note compares coherent sampling with window sampling and provides an explanation of how to evaluate high-speed analog-to-digital converters (ADCs) using either method, while detailing the advantages and disadvantages of each.

- Application note 3190, "Coherent Sampling Calculator (CSC)"
- Coherent Sampling Calculator (XLS, 81K)

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Performing an FFT assumes that a waveform is continuously sampled from -∞ to +∞. If one of the above mentioned conditions for coherent sampling is not met, non-coherent sampling

Windowing the input data is equivalent to convolving the spectrum of the original signal with the spectrum of the window. Although it is often assumed that no window is used for coherent sampling, in actuality, the input signal is convolved with a rectangular-shaped window of uniform height.

The frequency characteristic of a window is a continuous spectrum consisting of a main lobe and several side lobes. The main lobe is centered at the frequency of the input signal. Side lobes approach zero progressing from each side of the main lobe. An FFT, on the other hand, produces a discrete frequency spectrum. The continuous, periodic spectrum of a window is sampled by the FFT, just as an ADC would sample an input signal in the time domain. What appears in each frequency line of the FFT is the value of the continuous convolved spectrum at each FFT frequency component.

If the frequency components of the original signal match a frequency line exactly, as is the case when acquiring an integer number of cycles, only the main lobe of the spectrum can be seen. Side-lobes will not appear since the window spectrum approaches zero at bin-frequency intervals on either side of the main lobe. If a time record does not contain an integer number of cycles, the continuous spectrum of the window is shifted from the main lobe center by a fraction of the frequency bin. This corresponds to the difference between the frequency component and the frequency lines in the FFT spectrum. This shift causes side-lobes to appear in the spectrum. Therefore, the side-lobe characteristics of a given window directly affect the extent to which adjacent frequency components "leak into" neighboring frequency bins.

- The largest side-lobe level in decibels, with respect to the main lobe peak.
- Side-lobe roll-off, which is defined as the asymptotic decay rate of the side-lobe peaks.

Window Type | -3dB Main-Lobe Width | -6dB Main-Lobe Width | Maximum Side-Lobe Level | Side-Lobe Roll-Off Rate |

No Window (Rectangular) | 0.89 bins | 1.21 bins | -13.2dB | 20dB/decade 6dB/octave |

Hamming | 1.3 bins | 1.81 bins | -41.9dB | 20dB/decade 6dB/octave |

Hanning | 1.44 bins | 2 bins | -31.6dB | 60dB/decade 18dB/octave |

Blackman | 1.68 bins | 2.35 bins | -58dB | 60dB/decade 18dB/octave |

Different windows support different applications, and choosing the right one is not an easy task. Given that the signal contains strong interfering frequency components distant from the frequency of interest, a window with a high roll-off rate (e.g., a Hanning window) for the side-lobes should be chosen. However, if strong interfering signals are close to the frequency of interest, a window with a rather small maximum side-lobe level (e.g., a Flat Top window) is the more suitable choice. A waveform with adjacent components of identical magnitude is analyzed best when left within a rectangular window or no window at all. For a single-tone test in which the focus is on amplitude accuracy rather than its precise location in the frequency bin, a window with a wide main lobe (e.g., a Blackman window) is recommended.

Notes:

- N
_{WINDOW}must be a power of 2 to allow the use of a radix - FFT analysis.
- An irreducible ratio ensures identical code sequences not to be repeated multiple times. Unnecessary repetition of the same code is not desirable as it increases ADC test time.
- ADCs are usually characterized for and tested with sinusoidal input signals. Non-coherent sampling for sinusoidal input signals means that the first and the last sample of the input sinusoid are discontinuous with one another.
- Waveform discontinuance describes an input signal, for which an integer number of its cycles do not fit into a predefined window.