Problem: Consider a very simple example:
$y_k=x+v_k$
where $x$ is a deterministic unknown value. Here $x$ has the real value as 1. $y_k$ is a measurement. Our goal is to estimate the $x$ using measurement $y_k$.

This problem is very simple. It may arise when we try to meansure the length of an object. Or we can treat the measurements as a noisy signal.

State Space Model: In order to apply Kalman Filter. Rewrite it to state space model.
$x_{k}=1\cdot x_{k-1}+0$
$y_k=1\cdot x_{k}+v_i$
where $E(v_iv_j)=\delta_{ij}R$.

Apply Kalman Filter: then we have
$P_{k,k-1}=P_{k-1}+0$
$K_k=P_{k,k-1}(P_{k,k-1}+R)^{-1}$
$\hat{x}_{k,k-1}=\hat{x}_{k-1}$
$\hat{x}_k=\hat{x}_{k,k-1}+K_k(z_k-\hat{x}_{k,k-1})$
$P_k=(1-K_k)P_{k,k-1}(1-K_k)+K_kRK_k$
Choose initial values of $\hat{x}_0, P_0, R$ as $-1, 2, 3$.

Matlab Code:

clc;clear;close all
num=100;
x=ones(1,num); % real value
z=x;
for i=1:num
z(i)=x(i)+randn(1); % measurement
end
x_hat=sum(z)/num;

figure;
subplot(2,1,1);hold on;grid on
title 'estimation of x_k';
subplot(2,1,2); hold on; grid on
title 'estimation of P_k';
% initial value
x_k_1=-1;
P_k_1=2;
subplot(2,1,1);
plot(0,x_k_1, 'r>');
subplot(2,1,2);
plot(0,P_k_1, 'o');
R=3;
for k=1:num
z_k=z(k);

%P_k_k_1=P_k_1 % 1
K_k=P_k_1*inv(P_k_1+R); % 2
% x_k_k_1=x_k_1 % 3
x_k=x_k_1+K_k*(z_k-x_k_1); % 4
P_k=(1-K_k)*P_k_1*(1-K_k)'+K_k*R*K_k'; % 5

subplot(2,1,1)
plot(k,x_k,'r>');
plot(k,z_k,'g*');
subplot(2,1,2)
plot(k,P_k,'o');

% for next loop
P_k_1=P_k;
x_k_1=x_k;
end

Simulation results: