Transfer function of an ideal theoretical distorsionless surface acoustic wave filters

This paper presents the transfer function of the ideal theoretical distorsionless surface acoustic wave (SAW) bandpass filters, having a frequency bandwidth 2B. The required transfer function have constant magnitude and linear phase, centered at the carrier frequency f_c . In the time domain, the impulse response of the required SAW filter is a cos(2πf_ct) carrier wave, modulated by a sine function centered at the constant time delay t₀ of the transmission signals. This impulse response is multiplied by the rectangular window function to be a finite impulse response (FIR). Two methods are introduced. The first method, the Fourier transformation, is based on the convolution theorem principle of the required transfer function and the convolving weighted sine function of the rectangular window function. The second method uses the pre-envelope, complex envelope, and the conjugate theorem principle of the Fourier transformation. The nonflat response in the passband (Fresnel ripples), sidelobe rejection level in the stopband (Gibbs ripples) and transition bandwidth are discussed. Verification of the solution has been done and comparison between the two methods are also discussed. Typical values of the SAW filters are a carrier frequency f_c, of 100, 240, 300 MHz, bandwidth 2B of 46.7, 140, 120 MHz, and transducer length τ of 1.05 μs, 350 ns, 6 μs, respectively. The second method shows a very important criterion should be taken into design consideration which is EXP(±j2πf_ct₀) equal to unity, causing the product of the carrier frequency f_c and the transmission time delay t₀ (f_ct₀) should be an integer number