Introduction to signals - Continuous and Discrete Time Signals and Systems - page 43

Table 1.4. Values of the signal q 2 [ k ] for −10 ≤ k ≤ k

k − 10 − 9 − 8 − 7 − 6 − 5 − 4

q 2 [ k ] x [ − 5] = 0 0.1 x [ − 4] = 0 . 2 0.3 x [ − 3] = 0 . 4 0.5 x [ − 2] = 0 . 6

k − 3 − 2 − 1

q 2 [ k ] 0.7 x [ − 1] = 0 . 8 0.9 x [0] = 1 0.9 x [1] = 0.8 0.7

k 1 0

q 2 [ k ] x [2] = 0 . 6 0.5 x [3] = 0 . 4 0.3 x [4] = 0 . 2 0.1 x [5] = 0

expansion, the value of x [ k ]at k = 0 is retained in q [ k ]. The even-numbered
samples, where k is a multiple of 2, of q [ k ] equal x [ k / 2]. The odd-numbered
samples in q [ k ] are set to zero.

While determining the interpolated sequence x [ mk ], Eq. (1.54) inserts ( m − 1)
zeros in between adjacent samples of the DT sequence x [ k ], where x [ k ] is not
defined. Instead of inserting zeros, we can possibly interpolate the undefined
values from the neighboring samples where x [ k ] is defined. Using linear inter-
polation, an interpolated sequence can be obtained using the following equation:

k
m

x

if k is a multiple of integer m

x ( m ) [ k ] =

k
m

k
m

(1 − α ) x + α x otherwise,

(1.55)

k
m

k
m

where denotes

the nearest integer greater than or equal to ( k / m ), and α = ( k mod m ) / m . Note
that mod is the modulo operator that calculates the remainder of the division
k / m .For m

denotes the nearest integer less than or equal to ( k / m ),

= 2, Eq. (1.55) simplifies to the following:

k
2

x

if k is even

x (2) [ k ] =

k − 1
2

k + 1
2

0 . 5 x + x if k is odd.

Although, Eq. (1.55) is useful in many applications, we will use Eq. (1.54) to
denote an interpolated sequence throughout the topic unless explicitly stated
otherwise.

Example 1.18
Repeat Example 1.17 to obtain the interpolated sequence q 2 [ k ] = x [ k / 2] using
the alternative definition given by Eq. (1.55).

Solution
The non-zero values of q 2 [ k ] = x [ k / 2] are shown in Table 1.4, where the val-
ues of the odd-numbered samples of q 2 [ k ], highlighted with the gray back-
ground, are obtained by taking the average of the values of the two neighboring