FIR (matched) filter

For information on designing the digital matched filter using the graphical user interface, see Burst spectra and AFC plot (Ps).

The legacy bandpass design procedure computes two sets of filter coefficients, $f n i$ and $f n q$, such that the instantaneous quadrature samples at a given bin are:

$I = ∑ N = 0 N − 1 f n i × p n$, $Q = ∑ N = 0 N − 1 f n q × p n$

where N is the length of the filter. The input samples pn are centered on the range bin to which the (I, Q) pair is assigned. Note that some pn may overlap among adjacent bins. That is, the filter length may be greater than the bin spacing. The overlap introduces a slight correlation between successive bins, but the longer length allows a better filter to be designed. In the situation where the filter length design is smaller than the bin spacing, remaining samples pn are skipped until the start of the next range bin. In the legacy bandpass filter, Vaisala generally recommends the filter length be about 1.5 x the pulse length.

The sums above for I and Q are computed on RVP10 using a flexible FGPA that can perform billions of sums of products per second.

Reference phase

The reference phase for each transmitted pulse is computed using the same two FIR sums, except that bn is substituted for pn and choices of window function are either Rectangular, Hamming, or Blackman.

For magnetron systems, the N bn samples are centered on the transmitted burst.

For Klystron systems, the N bn samples may be obtained from the burst pulse (recommended) or from the CW STALO. If the Klystron is phase modulated by an external phase shifter (instead of the IFDR digital transmitter board), the samples must be from the modulated STALO.

Coefficients

The $f n i$ coefficients are computed as:

where f_IF is the radar intermediate frequency, f_SAMP is the IFDR sampling frequency, and l_n represents the coefficients of an N-point symmetric low-pass FIR filter that is matched to the bandwidth of the transmitted pulse. The multiplication of the l_n terms by the sin() terms effectively converts the low-pass filter to a band-pass filter centered at the radar IF.

The formula for the $f n q$ coefficients is identical except that sin() is replaced with cos().

The phase of the sinusoid terms, and the symmetry of the ln terms, has been chosen to have a valuable overall symmetry property when n is replaced with (N-1)-n, that is, the sequence is reversed:

The coefficients needed to compute I are the reverse of the i coefficients needed to compute Q. That is, if you know fⁱ _n, then you also know f^q_n.

When selecting to use the idealized matching filter, the fⁱcoefficients are computed similarly, except that the symmetric low-pass FIR filter ln is replaced by a scaling function so the gain/loss at the center frequency is 0 dB, and W[n] and windowing function.

Windowing functions W[n] are a signal shaping technique by increasingly reducing the amplitudes of samples from the center to the edge of the time domain. They may be used to increase signal coherency, which improves ground clutter suppression, or to increase the differentiation of 1st trip from multi-trip echo or from other types of interference. However, applying a windowing function will also generally decrease sensitivity and increase the variability seen in the final data types from the signal processor. For more information on windowing functions, see Frequency Domain Processing- Doppler power spectrum.

The types of windowing functions supported within the matched filter are Rectangular (no window), Triangular, Hann, Hamming, Blackman, Blackman Exact, Tukey, Kaiser, and Flat top. For any given noise bandwidth, the Rectangular window function provides the lowest root mean square error of the data estimates, and the Triangular window provides the best signal coherency.

The following chart shows windowing function properties matched to a 2.0 µs burst pulse. The Doppler spectrum is that of a nearby point clutter target after the application of each window.

Rectangular Filter loss 1.08 dB BW 0.368 MHz NEBW 0.416 MHz Noise sample -84.54 (H) -83.49 (V) MDS (Noise - Filter loss) -83.46 Clutter rejection 49 dB brstEg 0.145 nJ fltBrstEg 0.114 nJ fltMtch 0.13 dB

Triangular Filter loss 0.83 dB BW 0.530 MHz NEBW 0.555 MHz Noise sample -83.68 (H) -83.29 (V) MDS (Noise - Filter loss) -82.85 Clutter rejection 51 dB brstEg 0.145 nJ fltBrstEg 0.118 nJ fltMtch 0.11 dB

Hann Filter loss 0.76 dB BW 0.598 MHz NEBW 0.624 MHz Noise sample -83.18 (H) -82.80 (V) MDS (Noise - Filter loss) -82.42 Clutter rejection 50 dB brstEg 0.145 nJ fltBrstEg 0.122 nJ fltMtch 0.08 dB

Hamming Filter loss 0.79 dB BW 0.541 MHz NEBW 0.567 MHz Noise sample -83.55 (H) -83.16 (V) MDS (Noise - Filter loss) -82.76 Clutter rejection 50.5 dB brstEg 0.145 nJ fltBrstEg 0.121 nJ fltMtch 0.08 dB

Blackman Filter loss 0.67 dB BW 0.603 MHz NEBW 0.718 MHz Noise sample -82.62 (H) -82.13 (V) MDS (Noise - Filter loss) -81.95 Clutter rejection 49 dB brstEg 0.145 nJ fltBrstEg 0.125 nJ fltMtch 0.05 dB

Blackman Exact Filter loss -0.85 dB BW -0.535 MHz NEBW -0.577 MHz Noise sample -83.69 (H) -83.25 (V) MDS (Noise - Filter loss) -82.84 Clutter rejection 48 dB brstEg 0.145 nJ fltBrstEg 0.120 nJ fltMtch 0.07 dB

Tukey Filter loss 0.73 dB BW 0.598 MHz NEBW 0.624 MHz Noise sample -83.18 (H) -82.82 (V) MDS (Noise - Filter loss) -82.45 Clutter rejection 48.5 dB brstEg 0.145 nJ fltBrstEg 0.122 nJ fltMtch 0.07 dB

Kaiser Filter loss 0.59 dB BW 0.770 MHz NEBW 0.813 MHz Noise sample -82.10 (H) -81.68 (V) MDS (Noise - Filter loss) -81.58 Clutter rejection 49 dB brstEg 0.145 nJ fltBrstEg 0.125 nJ fltMtch 0.07 dB