basically of segments of the transfer function of ES and linear 
interpolation for the sample spacings Ax . , 2 Ax . , 4 AX , 
min min m 
in 
se = ar AX ain (indicated by the dashed curves in figure 2). 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
IH 
1 IM 
0.9 x Ps 
SF TN 
9.8 X \ M 
=] =, 
0.7 ER M 
* \ X | N 
0.6 : e X 
N 
ie We e 
* 1 X . 
\ N X : Transfer function 
9. ^d \ \ | LN for equispaced sampling 
0.3 X e apd linear interpolation 
{ N 
| 
0.2 1 
| 
0.1 + 
| 
| 
8 1 - T 1 v 
Q 0.1 0.2 9.3 0.4 045 
  
  
  
  
Figure 2: H(v,T,r) for sinusoidal input to PS and linear interpolation 
The parameters chosen for figure 2 are Ax in 1, threshold T - 0.5, and 
number of densification runs r = 2. Increasing the number of densifica- 
tion runs, e.g., tor = 4 and keeping Ax in = 1 (i.e., starting with a 
zero-sample spacing of 16 instead of 4) would add two more segments to 
the curve H( v,T,r) at the low frequency end (compare figure 4). This 
means a reduction in fidelity of reconstructing low frequency sinewaves; 
thus larger errors would result for sinewaves of long periods. The ex- 
tent to which fidelity is reduced depends on the threshold value 
(compare figure 2 with figure 4; in figure 2, where T = 0.5 was used, 
the fidelity does not drop below 95% at the low frequency end; in figure 
4, where T = 2, fidelity drops to approximately 80%) . 
Using a threshold T = 0 yields a function H(v, T,r) which is equal to 
the transfer function of ES except at Vv, = {1/4, 1/8, 3/8), if r = 2. 
For these critical frequencies, H( v,T,r) is zero. As illustrated in 
figure 3 for v - 1/4 and Ax z-4 (i.e., AX nin =-1,r = 2), the second 
differences in the zero-sampling-run are zero; therefore no further den- 
sification is done, although it would be necessary in order to allow at 
least partial reconstruction of the sinusoid. Inherent in the coarse to 
fine principle of PS is the risk that the constellation of the coarse 
sample prevents the necessary finer sampling. 
- 657 =