Periodicity
hI  t  
N / 2T
H  f  exp   j 2 ft  df (1.1)

 N / 2T
h t  
1 N / 21
t

H  f m  exp   j 2 m

T m  N / 2
T


 (1.2)

Equation (1.1) and (1.2) differ in the result of t t + T. Equation (1.2) is periodic, while 1.1 is not. The
difference is due to the fact that intermediate frequencies are inclued in (1.1). Following the Nyquist
method form S(f), define an set of N frequency points for T = MT
S  fk   M
s  ti  
1
T
N  / 2 1

m  N  / 2
N / 2 1

k  N  / 2
 k , M m (1.3)
S  f k  exp   j 2 f k ti  
1 N / 21
m  N


exp   j 2 ti  

T m N / 2
T  T

1 N / 21 N / 21
k


 k , Mm exp   j 2
ti 


T m  N / 2 k  N / 2
M T 

M / 2 1

k  M / 2
 i ,kN

  t
k 
i
(1.4)
 kT 
Then
m
M T
1 N  / 21
t

h t  
H  f m  S  f m  exp   j 2 m 

T m N / 2
T

fm 
(1.5)
Using the convolution theorem this becomes
h t  
N  / 2 1

i  N  / 2
hI  ti  s  t  ti  
M / 2 1

k  M / 2
hI  t  kT  (1.6)
Integral approximation
h t 
t
t
t



exp   j N   exp  j N  sin   N 
T
T
T



(1.7)
exp   j 2 ft  df 


2 j t
t
N / 2T
N / 2T
There is an obvious flaw in this. It is not periodic with time T. As T becomes very large with
fixed t Error! Reference source not found. becomes the integral result (1.7).
Figure 1 Periodic Error! Reference source not found. and non periodic results (1.7) for N=64
and Time=7 seconds. The periodic result in blue goes to zero faster than the non-periodic
result in black.
Figure 2 Non Peridic minus Periodic versus t/T
Equation (1.7) can be made periodic by adding to it all shifts by T, so that
 N

sin    t  kT  
 T

h t   
  t  kT 
k 

`
Figure 3 The nper was modified by adding periodic repeats of (1.7)up for up to +- 100T –
Core for 1000 in NPERMPER.ZIP
The difference above is linearly proportional to the number of repeats added as is appropriate for
1/T convergence.
In the vicinity of t=T/2 this oscillates about 2/(T) = 0.637/T. At the second maximum t=Ft=2.5 and the
value is F/(2.5)=2086. The very long range nature of h implies that t will not always be small compared to
T.