对此可以用滤波的方法将大的趋势项去掉。
测试的代码如下
% 测试积分对正弦信号的作用
clc
clear
close all
%% 原始正弦信号
ts = 0.001;
fs = 1/ts;
t = 0:ts:1000*ts;
f = 50;
dis = sin(2*pi*f*t); % 位移
vel = 2*pi*f.*cos(2*pi*f*t); % 速度
acc = -(2*pi*f).^2.*sin(2*pi*f*t); % 加速度
% 多个正弦波的测试
% f1 = 400;
% dis1 = sin(2*pi*f1*t); % 位移
% vel1 = 2*pi*f1.*cos(2*pi*f1*t); % 速度
% acc1 = -(2*pi*f1).^2.*sin(2*pi*f1*t); % 加速度
% dis = dis + dis1;
% vel = vel + vel1;
% acc = acc + acc1;
%
结：频域积分正常恢复信号，时域积分恢复加入的高频信息有误差
% 加噪声测试
acc = acc + (2*pi*f).^2*0.2*randn(size(acc));
% 结：噪声会使积分结果产生大的趋势项
figure
ax(1) = subplot(311);
plot(t, dis), title('位移')
ax(2) = subplot(312);
plot(t, vel), title('速度')
ax(3) = subplot(313);
plot(t, acc), title('加速度')
linkaxes(ax, 'x');
% 由加速度信号积分算位移
[disint, velint] = IntFcn(acc, t, ts, 2);
axes(ax(2)); hold on
plot(t, velint, 'r'), legend({'原始信号',
'恢复信号'})
axes(ax(1)); hold
on
plot(t, disint, 'r'), legend({'原始信号', '恢复信号'})
%% 测试积分算子的频率特性
n = 30;
amp = zeros(n, 1);
f = [5:30 40:10:480];
figure
for i = 1:length(f)
fi = f(i);
acc = -(2*pi*fi).^2.*sin(2*pi*fi*t); % 加速度
[disint, velint] = IntFcn(acc, t, ts, 2); % 积分算位移
amp(i) = sqrt(sum(disint.^2))/sqrt(sum(dis.^2));
plot(t, disint)
drawnow
% pause
end
close
figure
plot(f, amp)
title('位移积分的频率特性曲线')
xlabel('f')
ylabel('单位正弦波的积分位移幅值')
以上代码中使用IntFcn函数实现积分，它是封装之后的函数，可以实现时域积分和频域积分，其代码如下
%
积分操作由加速度求位移，可选时域积分和频域积分
function [disint, velint] = IntFcn(acc, t, ts, flag)
if flag == 1
% 时域积分
[disint, velint] = IntFcn_Time(t, acc);
velenergy = sqrt(sum(velint.^2));
velint = detrend(velint);
velreenergy = sqrt(sum(velint.^2));
velint =
velint/velreenergy*velenergy; disenergy = sqrt(sum(disint.^2));
disint = detrend(disint);
disreenergy = sqrt(sum(disint.^2));
disint = disint/disreenergy*disenergy; %
此操作是为了弥补去趋势时能量的损失
% 去除位移中的二次项
p = polyfit(t, disint, 2);
disint = disint - polyval(p, t);
else
% 频域积分
velint = iomega(acc, ts, 3, 2);
velint = detrend(velint);
disint = iomega(acc, ts, 3, 1);
% 去除位移中的二次项
p = polyfit(t, disint, 2);
disint = disint - polyval(p, t);
end
end
其中时域积分的子函数如下
% 时域内梯形积分
function [xn, vn] = IntFcn_Time(t, an)
vn = cumtrapz(t, an);
vn = vn - repmat(mean(vn), size(vn,1), 1);
xn = cumtrapz(t, vn);
xn = xn - repmat(mean(xn), size(xn,1), 1);
end
频域积分的子函数如下(此代码是一个老外编的，在频域内实现积分和微分操作)
function dataout = iomega(datain, dt,
datain_type, dataout_type)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% IOMEGA is a
MATLAB script for converting displacement, velocity, or
% acceleration
time-series to either displacement, velocity, or
% acceleration
times-series. The script takes an array of waveform data
% (datain),
transforms into the frequency-domain in order to more easily
% convert into
desired output form, and then converts back into the time
% domain
resulting in output (dataout) that is converted into the
desired
% form.
%
% Variables:
% ----------
%
% datain = input waveform
data of type datain_type
%
% dataout = output waveform
data of type dataout_type
%
% dt = time increment
(units of seconds per sample)
%
% 1 - Displacement
% datain_type = 2 -
Velocity
% 3 - Acceleration
%
% 1 - Displacement
% dataout_type
= 2 -
Velocity
% 3 - Acceleration
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Make sure that
datain_type and dataout_type are either 1, 2 or 3
if (datain_type < 1 || datain_type >
3)
error('Value for datain_type must be a 1, 2 or 3');
elseif (dataout_type < 1 || dataout_type
> 3)
error('Value for dataout_type must be a 1, 2 or 3');
end
% Determine
Number of points (next power of 2), frequency increment
% and Nyquist
frequency
N = 2^nextpow2(max(size(datain)));
df = 1/(N*dt);
Nyq = 1/(2*dt);
% Save frequency
array
iomega_array = 1i*2*pi*(-Nyq : df : Nyq-df);
iomega_exp = dataout_type - datain_type;
% Pad datain
array with zeros (if needed)
size1 = size(datain,1);
size2 = size(datain,2);
if (N-size1 ~= 0 && N-size2 ~=
0)
if size1 > size2
datain = vertcat(datain,zeros(N-size1,1));
else
datain = horzcat(datain,zeros(1,N-size2));
end
end
% Transform
datain into frequency domain via FFT and shift output (A)
% so that
zero-frequency amplitude is in the middle of the array
% (instead of the
beginning)
A = fft(datain);
A = fftshift(A);
% Convert datain
of type datain_type to type dataout_type
for j = 1 : N
if iomega_array(j) ~= 0
A(j) = A(j) * (iomega_array(j) ^ iomega_exp);
else
A(j) = complex(0.0,0.0);
end
end
% Shift new
frequency-amplitude array back to MATLAB format and
% transform back
into the time domain via the inverse FFT.
A = ifftshift(A);
datain = ifft(A);
% Remove zeros
that were added to datain in order to pad to next
% biggerst power
of 2 and return dataout.
if size1 > size2
dataout = real(datain(1:size1,size2));
else
dataout = real(datain(size1,1:size2));
end
return