Digital Filters - Digital Signal Processing: A Practitioner's Approach - page 53

3.3.3 Numerically Controlled Oscillator
Consider this statement:

For a sequence of numbers, multiply the previous number by 1.8 and subtract the
previous-previous number to result in a new number. Start with a previous number of 1
and a previous-previous number of 0.

Translated into an equation, this is

y k ¼ 1 : 8y k 1 y k 2 given y 0 ¼ 1 y 1 ¼ 0 : ð 3 : 16 Þ

Equation (3.16) results in the infinite sequence of Figure 3.9; it is a sinusoidal
wave with a normalised frequency f n ¼

1
2

cos 1 1 : 2 ¼ 0

: 0718. The infinite

3

2

1

0

−1

−2

−3

0 2 4 6 8 10 12 14 16 18 20

Figure 3.9 Numerically controlled oscillator

sequence f y k g generated by the 3-tuple f 1 8, 1, 0 g illustrates a method for data

compression. In addition, by varying only the coefficient of y k 1 , we can control the
frequency; this is an important result used in the implementation of a numerically
controlled oscillator (NCO). Consider a simple pendulum oscillating. If we watch
this using a stroboscope, the angle seen at the time of flash is

:

k ¼ k 1 k 2 ,

where is a constant that depends on the flash frequency, the length of the

pendulum, and the acceleration due to gravity at that place.

3.4 Unity-Gain Narrowband Filter
One of the most popular filters is the narrowband filter (NBF). It is used in every
domain of filter design. All filter designs of higher order use cascades of NBFs
having different parameters. In superheterodyning receivers, the NBF is used
immediately after the mixer stage. It is also very common in notch filters. In
such filters there is a need to have unity gain around the peak response and zero