Digital Architectures for UWB Beamforming Using 2D IIR Spatio-Temporal Frequency-Planar Filters

Table 1

Filter design equations showing algebraically defined filter coefficients as functions of LC ladder prototype network parameters and steered angle tan𝜓=sin𝜃. Here, ⋅ refers to dot product between vectors.

Order; of filter	Arbitrary coefficients	Coefficients
	𝑒𝑚	𝑏𝑖𝑗
1		𝑏𝑖𝑗=𝑒̂⋅𝛽𝑖𝑗,𝑏00=0
	𝐿𝑖=𝐿cos𝜓,	𝑒̂=(𝑒1,𝑒2,𝑒3)/𝜖
	𝐿𝑗=𝐿sin𝜓,	𝜖=𝑒1+𝑒2+𝑒3
	𝑒1=1+𝑅𝑠,	𝛽01=(1,1,−1)
	𝑒2=𝐿𝑖,𝑒3=𝐿𝑗	𝛽10=(1,−1,1)
		𝛽11=(1,−1,−1)
2		𝑏𝑖𝑗=𝑒̂⋅𝛽𝑖𝑗,𝑏00=0
	𝐿𝑖=𝐿1cos𝜓,	𝑒̂=(𝑒1,𝑒2,𝑒3,𝑒4,𝑒5,𝑒6)/𝜖
	𝐿𝑗=𝐿1sin𝜓,	𝜖=𝑒1+𝑒2+𝑒3+𝑒4+𝑒4+𝑒6
	𝐶𝑖=𝐶2cos𝜓,	𝛽01=(2,2,0,2,−2,0)
	𝐶𝑗=𝐶2sin𝜓.	𝛽02=(1,1,−1,1,1,−1)
	𝑒1=1+𝑅𝑠,	𝛽10=(2,0,2,−2,2,0)
	𝑒2=𝐿𝑖+𝑅𝑠𝐶𝑖,	𝛽11=(4,0,0,−4,−4,0)
	𝑒3=𝐿𝑗+𝑅𝑠𝐶𝑗,	𝛽12=(2,0,−2,−2,2,0)
	𝑒4=𝐿𝑖𝐶𝑖,𝑒5=𝐿𝑗𝐶𝑗,	𝛽20=(1,−1,1,1,1,−1)
	𝑒6=𝐿𝑖𝐶𝑗+𝐿𝑗𝐶𝑖.	𝛽21=(2,−2,0,2,−2,0)
		𝛽22=(1,−1,−1,1,1,1)
3		𝑏𝑖𝑗=𝑒̂⋅𝛽𝑖𝑗,𝑏00=0,
		𝑒̂=(𝑒1,𝑒2,𝑒3,𝑒4,𝑒5,𝑒6,𝑒7,𝑒8,𝑒9,𝑒10)/𝜖,
	𝐿𝑖=𝐿1cos𝜓,	𝜖=𝑒1+𝑒2+𝑒3+𝑒4+𝑒5+𝑒6+𝑒7+𝑒8+𝑒9+𝑒10
	𝐿𝑗=𝐿1sin𝜓,	𝛽01=(3,3,1,3,−1,1,3,−3,1,−1)
	𝐶𝑖=𝐶2cos𝜓,	𝛽02=(3,3,−1,3,−1,−1,3,3,−1,−1)
	𝐶𝑗=𝐶2sin𝜓,	𝛽03=(1,1,−1,1,1,−1,1,−1,−1,1)
	𝐿𝑘=𝐿3cos𝜓,	𝛽10=(3,1,3,−1,3,1,−3,3,−1,1)
	𝐿𝑙=𝐿3sin𝜓,	𝛽11=(9,3,3,−3,−3,1,−9,−9,−1,−1)
	𝑒1=1+𝑅𝑠,	𝛽12=(9,3,−3,−3,−3,−1,−9,9,1,−1)
	𝑒2=𝐿𝑖+𝐿𝑘+𝑅𝑠𝐶𝑖,	𝛽13=(3,1,−3,−1,3,−1,−3,−3,1,1)
	𝑒3=𝐿𝑗+𝐿𝑙+𝑅𝑠𝐶𝑗,	𝛽20=(3,−1,3,−1,3,−1,3,3,−1,−1)
	𝑒4=𝐿𝑖𝐶𝑖+𝐿𝑘𝐶𝑖,	𝛽21=(9,−3,3,−3,−3,−1,9,−9,−1,1)
	𝑒5=𝐿𝑗𝐶𝑗+𝐿𝑙𝐶𝑗,	𝛽22=(9,−3,−3,−3,−3,1,9,9,1,1)
	𝑒6=𝐿𝑖𝐶𝑗+𝐿𝑗𝐶𝑖+𝐿𝑘𝐶𝑗+𝐿𝑙𝐶𝑖,	𝛽23=(3,−1,−3,−1,3,1,3,−3,1,−1)
	𝑒7=𝐿𝑖𝐿𝑘𝐶𝑖,𝑒8=𝐿𝑗𝐿𝑙𝐶𝑗,	𝛽30=(1,−1,1,1,1,−1,−1,1,1,−1)
	𝑒9=𝐿𝑖𝐿𝑘𝐶𝑗+𝐿𝑖𝐿𝑙𝐶𝑖+𝐿𝑗𝐿𝑘𝐶𝑖,	𝛽31=(3,−3,1,3,−1,−1,−3,−3,1,1)
	𝑒10=𝐿𝑖𝐶𝑖𝐿𝑙+𝐿𝑗𝐿𝑘𝐶𝑗+𝐶𝑖𝐿𝑗𝐿𝑙.	𝛽32=(3,−3,−1,3,−1,1,−3,3,−1,1)
		𝛽33=(1,−1,−1,1,1,1,−1,−1,−1,−1)