Revisiting the Basics - Digital Signal Processing: A Practitioner's Approach - page 23

p . Engineers treat y k as an
ordered pair of sequences, popularly known as in-phase and quadrature sequences.
Note that there is no change in frequency but only change in amplitude and phase.
This concept is used in many practical systems, such as network analysers,
spectrum analysers and transfer function analysers.

y k as a complex sequence y k ¼ y i k þ jy k , where j ¼

2.1.3 Stability
Another important property defining the system is its stability. By definition if
u k ¼ k and the corresponding output is y k ¼ h k , and if the sequence h k is

k is the unit sample. 1 We can also

convergent, then the system is stable. Here

define a summation S ¼ P k ¼ 0 h k and if S is finite, then the system is stable. The
physical meaning is that the area under the discrete curve h k must be finite. There
are many methods available for testing convergence. One popular test is known as
the ratio test, which states that if j h k þ 1 = h k j <

1 ; then the sequence h k is convergent,

hence the system is stable.

2.1.4 Shift Invariance
Shift invariance is another important desirable property. A shift in the input results
in an equal shift in the output (u k N ¼) y k N ). This property is known as shift
invariance. Linear and shift invariant (LSI) systems have very interesting properties
and are mathematically tractable. This is the reason why many theoretical deriva-
tions assume this property.

2.1.5 Impulse Response
If the input u k is a unit sample then the system response y k is called a unit sample

response of the system, h k . This has a special significance and completely
characterises the system. This sequence could be finite or infinite. The sequence
h k is said to be finite if

k ;

h k ¼ finite for k < N

¼ 0 for k N : ð 2 : 4 Þ

If the impulse response h k is finite as defined in (2.4), then the system is a finite
impulse response (FIR) system and it has a special significance. Figure 2.3 shows a
typical unit sample response of a system defined by (2.13) and the unit sample
response for this system is infinite. Traditionally, systems in the discrete domain
originate from differential equations describing a given system, and these differ-
ential equations take the form of difference equations in the discrete domain.

1

Impulse response is used in continous systems.