LeCroy DDA 7 Zi series, Operator'S Manual

Page: 153 / 544

Operator's Manual
WP700Zi-OM-E-RevA
152
Spectral analysis theory assumes that the signal for transformation is of infinite duration. Since no physical signal can meet this condition, a
useful assumption for reconciling theory and practice is to view the signal as consisting of an infinite series of replicas of itself. These replicas
are multiplied by a rectangular window (the display grid) that is zero outside of the observation grid.
An FFT operation on an N-point time domain signal can be compared to passing the signal through a comb filter
consisting of a bank of N/2 filters. All the filters have the same shape and width and are centered at N/2 discrete
frequencies. Each filter collects the signal energy that falls into the immediate neighborhood of its center
frequency. Thus it can be said that there are N/2 frequency bins. The distance in Hz between the center
frequencies of two neighboring bins is always the same: Delta f.
Power (Density) Spectrum
Because of the linear scale used to show magnitudes, lower amplitude components are often hidden by larger components. In addition to the
functions offering magnitude and phase representations, the FFT option offers power density and power spectrum density functions. These
latter functions are even better suited for characterizing spectra. The power spectrum (V
2
) is the square of the magnitude spectrum (0 dBm
corresponds to voltage equivalent to 1 mW into 50 ohms.) This is the representation of choice for signals containing isolated peaks — periodic
signals, for instance.
The power density spectrum (V
2
/Hz) is the power spectrum divided by the equivalent noise bandwidth of the filter
associated with the FFT calculation. This is best employed for characterizing broadband signals such as noise.
Memory for FFT
The amount of acquisition memory available will determine the maximum range (Nyquist frequency) over which signal components can be
observed. Consider the problem of determining the length of the observation window and the size of the acquisition buffer if a Nyquist rate of
500 MHz and a resolution of 10 kHz are required. To obtain a resolution of 10 kHz, the acquisition time must be at least:
T = 1/Delta f = 1/10 kHz = 100 ms
For a digital oscilloscope with a memory of 100 kB, the highest frequency that can be analyzed is:
Delta f x N/2 = 10 kHz x 100 kB/2 = 500 MHz
FFT Pitfalls to Avoid
Take care to ensure that signals are correctly acquired: improper waveform positioning within the observation
window produces a distorted spectrum. The most common distortions can be traced to insufficient sampling, edge
discontinuities, windowing or the "picket fence" effect.
Because the FFT acts like a bank of band-pass filters centered at multiples of the frequency resolution,
components that are not exact multiples of that frequency will fall within two consecutive filters. This results in an
attenuation of the true amplitude of these components.
Picket Fence and Scallop
The highest point in the spectrum can be 3.92 dB lower when the source frequency is halfway between two discrete frequencies. This
variation in spectrum magnitude is the picket fence effect. The corresponding attenuation loss is referred to as scallop loss. LeCroy scopes
automatically correct for the scallop effect, ensuring that the magnitude of the spectra lines correspond to their true values in the time domain.
If a signal contains a frequency component above Nyquist, the spectrum will be aliased, meaning that the
frequencies will be folded back and spurious. Spotting aliased frequencies is often difficult, as the aliases may
ride on top of real harmonics. A simple way of checking is to modify the sample rate and observe whether the
frequency distribution changes.
Leakage
FFT assumes that the signal contained within the time grid is replicated endlessly outside the observation window. Therefore if the signal
contains discontinuities at its edges, pseudo-frequencies will appear in the spectral domain, distorting the real spectrum. When the start and
end phase of the signal differ, the signal frequency falls within two frequency cells, broadening the spectrum.
The broadening of the base, stretching out in many neighboring bins, is termed leakage. Cures for this are to
ensure that an integral number of periods is contained within the display grid or that no discontinuities appear at
the edges. Another is to use a window function to smooth the edges of the signal.
Choosing a Window
The choice of a spectral window is dictated by the signal’s characteristics. Weighting functions control the filter response shape, and affect
noise bandwidth as well as side lobe levels. Ideally, the main lobe should be as narrow and flat as possible to effectively discriminate all
spectral components, while all side lobes should be infinitely attenuated. The window type defines the bandwidth and shape of the equivalent
filter to be used in the FFT processing.
In the same way as one would choose a particular camera lens for taking a picture, some experimenting is generally necessary to determine
which window is most suitable. However, the following general guidelines should help.