Linear interpolation

This method is a two-point interpolation. We input the values of the nearest two points in the k_z domain, then use the following equation to get the value of a point between them:  

$$C_{n+ \delta n}= (1- \delta n)C_n +(\delta n) C_{n+1}$$ (3)

where $C_{n+ \delta n}$ is the value of the point to be interpolated, C_n is the value of the point on its left side, and C_n+1 is the value of the point on its right side. $\delta n$ is the value of the distance between the interpolated point and the sample point on its left side.

The linear interpolation in the frequency domain is equivalent to a convolution with a triangle function in the frequency domain Harlan (1982). In time-space domain, its counterpart is multiplication by a $sinc^2\frac{ t}{ T }$ function. The multiplication by the $sinc^2\frac{ t}{ T }$ causes the original values between T/2 and T to be weaker than the wraparound between -T/2 and . So in Figure 2, for which linear interpolation was used, the lower part of the section shows that the artifacts almost replace the original correct information.