A transformation for obtaining a linear phase variable cutoff lowpass digital fdter has been recently descnied in the literature [ 11. In this letter, this transformation is worked out for the case of linear phase bandpass digital mters with variable bandwidth and center frequency.In many applications of a digital bandpass filter, e.g., spectrum analysis. it is required to vary the bandwidth or the center frequency. In general, this can be achieved by changing all the coefficients of the digital filter. This is a cumbersome process particularly if the center frequency or bandwidth is to be varied quite frequently. It would be desirable to have a scheme wherein the variation can be effected through a single parameter in the structure. Oppenheim etal.[ 11 have suggested a transformation and given constraints on the control parameters A0 and A 1 for a linear phase low-pass variable cutoff filter. The transformation has been further examined in detail in [ 21. In this letter, the constraints on A0 and A1 are derived for the case of a linear phase bandpass filter with variable bandwidth or center frequency.Illustrative examples are given. Let the ( 2 N + 1) point zero-phase FIR prototype.have the frequency response n =O where b (n) are related to h (n), the impulse response coefficients of the prototype filter through Chebyshev polynomials. With the transformation [ 11 c o s w = A o + A l W S R ( 2 )on (l), a set of transformed filters is obtained where cutoff is controlled by A0 and A 1. For low pass filters it is desirable that the dc gain be invariant, giving the condition A0 + A 1 = 1. Transformation (2) then becomes a single parameter one. For bandpass filters the constraints on A0 and A can be derived as follows.Let the prototype linear phase bandpass filter and the desired transformed filter have symmetrical response about center frequency ( W O , n o ) and let w1, w2, and fZ1,. R2 be lower and upper cutoff frequencies of the prototype and transformed fdters, respectively. Then from (2), COS W I ,~ =A0 +A1 COS R1,2 (31, (4) so that A l = ( c o S W 1 -C O~~2 ) / ( c o~R l -c o~5 1 2 ) (5 1 = [sin ( A w /~) sin wo]/[sin (AR/2) sin n o ]where A X = X 2 -X 1 ~~~x~= ( x~+ x~) /~, x = w o I R .If the bandwidth is to be varied keeping the center frequency constant,simplifies to A = sin (Aw/2)/sin (AR/2). (8) Since, from (3), A0 and A are related by A o = c o s w l -A l c o s 5 2 1we conclude that only a single parameter A 1 can control the bandwidth of the bandpass filter, retaining, of course, its amplitude characteristics. The frequency response of the transformed filters obtained through (l), 24, 1978. n Fig. 1. (a) Frequency response of variable bandwidth, wide-band bandlower stopband edge = 0.15, lower passband edge = 0.25; upper passpass filter for various values of A I . Prototype specifications are: band edge = 0.40, upper stopband edge = 0.475; length = 33; error weighting = 1.0, 1.0, 1.0. (b) Frequency response of variable bandwidth narrowband bandpass filter for various values of A 1. Prototype specifica...
