Derivatives of Periodic Sampling - Digital Alias-free Signal Processing - page 170

1. For signal frequencies mf s − f a and n ≤ N / 2, it is found that

a a n

n

1)]

N
2

a r
N

sin(2 π f 0 T s n )
sin(2 π f 0 T s )

≤ = + cos[2 π f 0 T s ( n −

sin(2 π f 0 T s n )
sin(2 π f 0 T s )

1)]

b r
N

−

sin[2 π f 0 T s ( n

− ,

(8.12)

b a n

sin(2 π f 0 T s n )
sin(2 π f 0 T s )

1)]

N
2

a r
N

sin[2 π f 0 T s ( n

≤ = −

n

1)]

b r
N

sin(2 π f 0 T s n )
sin(2 π f 0 T s )

cos[2 π f 0 T s ( n

− − − . (8.13)

2. For signal frequencies mf s − f a and n > N / 2,

a n

N
2 +

α 1

sin( π f 0 T s N )
sin(2 π f 0 T s )

α 1

N
2

a r
N

sin( π f 0 T s N )
sin(2 π f 0 T s )

b r
N

> = cos − sin

n

3

1) m ( N + 1) a r
N

( − N

2

α 2
sin (2 π f 0 T s ) cos

sin

+ − + α

α 3

1) m ( N + 1) b r
N

( − sin α

2
sin(2 π f 0 T s ) sin

− , (8.14)

b a n

sin ( π f 0 T s N )
sin (2 π f 0 T s )

α 1

N
2

α 1

N
2

a r
N

b r
N

sin( π f 0 T s N )
sin(2 π f 0 T s )

> = − +

sin cos

α 3

1) m ( N + 1) a r
N

( − α 2

sin(2 π f 0 T s ) sin

sin

+

n

3

1) m ( N + 1) b r
N

( − N

2

sin α

2
sin(2 π f 0 T s ) cos

− − − α , (8.15)

where

2 π f 0 T s N
2

1 2 π f 0 T s n

N
2

α = − , α = − ,

1 2

2 π f 0 T s n

N
2

α 3 = + . (8.16)

Similar formulae can be derived for signal frequencies mf s + f a .