Ratio Mathematica Volume 46, 2023

Linear predictor and autocorrelation for
noisy and delayed digital signal

G.Vinu Priya*
Jothilakshmi R†

Abstract

This paper deals with the association between the linear prediction
and digital signal modeling and ends up with the suitable ways to pre-
dict the signal by considering a stationary signal yn. The linear pre-
diction of signal modeling based on the finite past and the solutions
are arrived in a recursive manner. Further we analyzed the wiener
filter along with spectral theorem and autocorrelation in terms ofpre-
dictive analysis. This estimates the gap function along with delay
and noise. The delayed signal’sproperties are analyzed like causal,
stability and applied these into optimum filtering. Finally the pre-
dicted error is compared with linear predictor and Wiener filter. Then
transfer function is applied to estimate the interval function and gap
function along with delay.

Keywords: Linear Predictor, Weiner filter, gapped function, de-
lay, autocorrelation, orthogonal.

2020 AMS subject classifications: 39 A10, 39 A45. 1

*PG and Research Department of Mathematics, D.K.M. College for Women (Autonomous),
India. e-mail:vinupriya14@gmail.com,

†PG and Research Department of Mathematics, Mazharul Uloom College, Tamil Nadu, India.
e-mail:jothilakshmiphd@gmail.com.
1Received on September 15, 2022. Accepted on December 15, 2022. Published on March 20,
2023. DOI: 10.23755/rm.v46i0.1083. ISSN: 1592-7415. eISSN: 2282-8214. ©G.Vinu Priya et
al. This paper is published under the CC-BY licence agreement.

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G.Vinu Priya and Jothilakshmi R

1 Introduction

Due to recent developments in Digital signal processing and communication
technology, and its subfields of spectrum estimation, real-time adaptive signal
processing and prediction algorithms comes into an attention of many researchers
Chen et al. [2006], Dogariu et al. [2021] and Gland and Oudjane [2003]. The uni-
fied extension of this digital signal processing is developed in terms of analysis
of geometrical point of view, linear estimator and applied various algorithms like
Gram-Schmidt orthogonalizations, lattice realizations and so on. Further these
concepts deals with the autoregressive extensions and singular autocorrelation
matrices and their sinusoidal representations Mao et al. [2017]. This motivates
us to proceed further with linear prediction Makhoul [1975] and Pituk [2004].

This paper is organized as follows. Section II focuses the linear prediction
and digital signal modeling Welch et al. [2006]. Section III applies the autore-
gressive models into the prediction coefficients. In section IV Linear Predictions
and Levinson’s Formula are applied in the random input signal. Finally, section V
concludes the paper.

2 Linear predictor

The linear prediction and digital signal modeling and ends up with the suitable
ways to predict the signal by considering a stationary signal yn. This rules the
signal pattern as follows

Syy(z) = σ
2
�B(z)B(z

−1)�n → (B(z)) → yn (1)

Here B(z) be any filter as bounded, �n be a sequence of noise term by spectral
factoring theorem. Let Ryy(k) be the autocorrelation of yn:

Ryy(k) = E[yn + kyn]

This is used to predict the present value through the past values by using Yn−1 =
{yi,−∞ < i ≤ n − 1}. If y1(n) = yn−1, then the linear prediction is identified
and compared with the optimum Wiener filtering and estimated the signal y1(n).
Now we identify Y1(z) = z−1Y (z) with the spectral value B(z). Now define the
optimum filter H(z) as

H(z) =
1

σ2�B(z)
[
σ2�B(z)B(z

−1)

B(z−1)
], (2)

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Linear predictor and autocorrelation for noisy and delayed digital signal

Here B(z) be a causal and stable filter, and extend the causal and stale filter as
zB(z) is then

zB(z) = z(b1z
−1 + b2z

−2 + b3z
−3 + · · ·)

The optimum filter H(z) is then

H(z) = z[1−
1

B(z)
] (3)

yn → (z−1) −→y1(n) (H(z)) → ŷn/n−1

This filter output is y1(n) and the consequent output is predicted by yn/n−1. The
predicted error is defined as �n. In the figure 1, the indicator line separates the

Figure 1: Error Predictor through Wiener Filter

linear predictor part and Wiener filter part in the signal error prediction [2, 7].
Apply the reduction equation (1) in terms of the predicted error filter A(z) as,

Syy(z) =
σ2�

A(z)A(z−1
(4)

and

A(z)Syy(z) =
σ2�

A(z−1)
(5)

this follows that

S�y(z) = A(z)Syy(z) (6)

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G.Vinu Priya and Jothilakshmi R

furthermore

R�y(k) = E[�nyn−k] =
∞∑
i=0

aiRyy(k − i) (7)

which is recognized by the interval function [3]. Now construct �n from the or-
thogonal complement of Yn−1 = yn−k,k = 1,2, . . ., and hence yn−k is orthogonal
to all k = 1,2, . . .. Therefore, the equation (7) implies

R�y(k) = E[�nyn−k] =
∞∑
i=0

aiRyy(k − i) = 0 (8)

This result follows from the z-domain equation of (6) and interval function. Ap-
plying the symmetry property in (7) provided k = 0 and we get

σ2� = E[�
2
n] = E[�nyn] = Ryy(0) + a1Ryy(1) + a2Ryy(2) + ... (9)

Combined the equations (8) and (9),

∞∑
i=0

aiRyy(k − i) = σ2�δ(k),k ≥ 0 (10)

This normal equation is extended with the parameters {a1,a2, . . . ,σ2�} based
on the output signal? yn and this is computed with Ryy(k).

3 Autoregressive models
In general, the prediction coefficients are infinite since the predictor is predi-

cated on the infinite past. When yn is autoregressive, then the signal model B(z)
is defined as

B(z) =
1

(1 + a1z−1 + a2z−2 + . . . + apz−p)
(11)

This shows that the prediction filter is polynomial

A(z) = 1 + a1z
−1 + a2z

−2 + . . . + apz
−p (12)

The output function yn is defined for uncorrelated sequence �n, we get

yn + a1yn−1 + a2yn−2 + ... + apyn−p = �n (13)

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Linear predictor and autocorrelation for noisy and delayed digital signal

further optimum prediction of yn is written like

ŷn/n−1 = −[a1yn−1 + a2yn−2 + ... + apyn−p] (14)

Here most effective prediction of yn is calculated based on the past p samples. The
infinite set of equations (10) or (11) remains valid and the primary p + 1 samples
coefficients {1,a1,a2, . . . ,ap}are nonzero [8, 14]. These primary past samples are
a part of the equation (11) and these samples are enough to define the parameters
of {a1,a2, . . . ,ap;σ2�}:


Ryy(0) Ryy(1) · · · Ryy(p)
Ryy(1) Ryy(0) · · · Ryy(p−1)
Ryy(2) Ryy(1) · · · Ryy(p−2)

...
...

...
...

Ryy(p) Ryy(p−1) · · · Ryy(0)


 =




1
a1
a2
...
ap


 =



σ2�
0
0
...
0


 (15)

These equations are solved efficiently through Levinson’s algorithm and this al-
gorithm needs O(p2) operations and O(p) memory locations. O(p3) and O(p2)
which is necessary to calculate the inverse of the autocorrelation matrix Ryy.
The parameters {a1,a2, . . . ,ap;σ2�} completely determines yn. By considering
z = ejω in the equation (5) we determine

Syy(ω) =
σ2�

|A(ω)|2
=

σ2�
|1 + a1e−jω + a2e−2jω + . . . + ape−jωp|

(16)

The normal equations (16) build is used to approximate and estimates the param-
eters {a1,a2, . . . ,ap;σ2�}. There are many various ways to extract the estimates
and the parameters. Here are the few methods

1. Yule-Walker methodology

2. Variance methodology and

3. Burg’s methodology.

Autocorrelations Ryy(k) of equation (16) is wrriten based on the Yule-Walker
methodology, is

Ryy(k) =
1

N

N−1−k∑
n=0

yn+kyn (17)

The primary p + 1 changes are required in (16) as like p ≤ N − 1 based on the
parameters {â1, â2, . . . , âp; σ̂2�}. This represents the block of N samples and filter

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G.Vinu Priya and Jothilakshmi R

parameters (i.e. p + 1). To synthesize the random samples, variance σ̂2� would be
generated and pass through the generator filter whose coefficients are calculated
like,

B̂(Z) =
1

Â(Z)
=

1

|1 + â1z−1 + â2z−2 + +âpz−p|2
(18)

4 Linear predictions and Levinson’s Formula
In this section, we come accross that if the autoregressive random input sig-

nal is of order p, then the optimum linear predictor reduces to a predictor of or-
der p. A geometrical method to perceive this property could be extended in to
the projection of yn onto the topological subspace based on the output signal
{yn−i,1 ≤ i < ∞} and the same could be reduced based on past samples; i.e.
{yn−i,1 ≤ i ≤ p}. This generates the output function yn.

Consider a stationary series (based on time) yn with the autocorrelation func-
tion R(k) = E[yn+kyn]. For any given p, the output function takes the following
new form Consider a stationary series (based on time) yn with the autocorrela-
tion function R(k) = E[yn+kyn]. For any given p, the output function takes the
following new form

ŷn = −[a1yn−1 + a2yn−2 + ... + apyn−p] (19)

The prediction coefficients are chosen to reduce the mean square error as

ε = E[e2n] (20)

where en is the predicted error and define en as follows

en = yn − ŷn = yn + a1yn−1 + a2yn−2 + ... + apyn−p (21)

E[enyn−i] = 0, (22)

By substituting (21) in the equation (22), we get p linear equations

p∑
j=0

ajE[yn−jyn−i] =

p∑
j=0

R(i− j)aj = 0 (23)

By (22), we found the reduced value as

σ2� = E[enyn] (24)

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Linear predictor and autocorrelation for noisy and delayed digital signal

Equations (23) and (24) may be combined into the matrix equation like (p + 1)×
(p + 1),


Ryy(0) Ryy(1) · · · Ryy(p)
Ryy(1) Ryy(0) · · · Ryy(p−1)
Ryy(2) Ryy(1) · · · Ryy(p−2)

...
...

. . .
...

Ryy(p) Ryy(p−1) · · · Ryy(0)


 =




1
a1
a2
...
ap


 =



σ2�
0
0
...
0


 (25)

which is identical for equation (16) for the autoregressive case.

It was necessary to connect the order of the predictor associate with the previ-
ous one. Hence the lower order optimum predictors also are calculated. Consider
the gap function as

gp(k) = E[(

p∑
i=0

apiyn−i)yn−k] =

p∑
i=0

apiR(k − i) (26)

Figure 2: Gap conditions for the delay

These gap conditions are an equivalent because of the orthogonal equations
(22) which is illustrated in figure 2. Utilizing gp(k) construct a new function with
space gp+1(k) from the past p + 1 hence we get, gp(k) → gp(−k). A delay of
(p + 1) time can realigned and illustrated in the following figure. This shows the
minimum of p and choosen the parameter γp+1 and gp+1(k) adds an additional
delay which deviates the length p + 1 are illustrated in the figure 3.

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G.Vinu Priya and Jothilakshmi R

Figure 3: Gap (delay) conditions for gp(k) and gp(p + 1−k)

5 Conclusions
In this paper, linear prediction is predicted the present value through the past

values. The linear prediction of signal modeling related to finite past and the so-
lutions are arrived in a recursive manner. Further we analyzed the wiener filter
along with spectral theorem and autocorrelation in terms of predictive analysis.
This estimates the gap function along with delay and noise. There will be an in-
dicator line which separates the linear predictor part and Wiener filter part in the
signal error prediction. This normal equation is extended with the signal param-
eters based on the output signal yn and this is computed with Ryy(k). Then the
infinite matrix equation is reduced to a finite form and, moreover, the Ryy(k) is
obviously measurable. Finally the predicted error is compared with linear pre-
dictor and Wiener filter. Then transfer function is applied to estimate the interval
function and gap function along with delay. Finally the gapped function gp(k) and
gp(p + 1−k) possess the same value.

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Linear predictor and autocorrelation for noisy and delayed digital signal

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