Table 1.

Frequency responses of the differencing schemes

Type of differencing		Fourier domain expression
D1	No subtraction	$Y_1(e^{j\omega })=Y\left(\begin{array}{c}j\omega \\ e\end{array}\right)$
D2	Pairwise subtraction	Y2(ejω) = 0.5·[ Y(ejω/2)(1-ejω/2) + Y(ej(ω+ 2π)/2)(1-ej(ω+2π)/2)]
D3	Running subtraction	Y3(ejω) = Y(ej(ω+π))(1-ej(ω+π))
D4	Surround subtraction	Y4(ejω) = Y(ej (ω+π))(− 1 + 2ej(ω+π) − e 2j(ω+π))
D5	Sinc subtraction	Y5(ejω) = Y2(ej 2ω)S(ejω )*

^*Y_2z = is the Fourier transform of y_2z[n], which is in turn the same as y₂[n] (the time domain output of the pairwise subtraction case) but zero filled between samples. In an ideal case (an Infinite Impulse Response filter), S would be a perfect rect function but, depending on the implementation, it is the Fourier transform of a truncated sinc function instead, and is dependent on the choice of truncated sinc.