<> First order low pass filter

<>1. First order continuous low pass filter

y ( s ) r ( s ) = a s + a \frac{y(s)}{r(s)}=\frac{a}{s+a} r(s)y(s)​=s+aa​

<>2. Convert to discrete form

Conversion to differential equation :
y ˙ ( t ) + a y ( t ) = a r ( t ) \dot{y}(t)+ay(t)=ar(t) y˙​(t)+ay(t)=ar(t)
The first order forward difference is used to separate and disperse :
y ( t ) ˙ = y [ ( k + 1 ) T ] − y ( k T ) T
\dot{y(t)}=\frac{y[(k+1)T]-y(kT)}{T}y(t)˙​=Ty[(k+1)T]−y(kT)​
That's it :
y [ ( k + 1 ) T ] − y ( k T ) T + a y ( k T ) = a r ( k T )
\frac{y[(k+1)T]-y(kT)}{T}+ay(kT)=ar(kT)Ty[(k+1)T]−y(kT)​+ay(kT)=ar(kT)

y [ ( k + 1 ) T ] − y ( k T ) + T a y ( k T ) = T a r ( k T ) y[(k+1)T] -
y(kT)+Tay(kT)=Tar(kT)y[(k+1)T]−y(kT)+Tay(kT)=Tar(kT)

y [ ( k + 1 ) T ] = ( 1 − a T ) y ( k T ) + T a r ( k T ) y[(k+1)T]
=(1-aT)y(kT)+Tar(kT)y[(k+1)T]=(1−aT)y(kT)+Tar(kT)

<>3. For example

Speed of differential calculation :
v ( k + 1 ) = ( 1 − a ) v ( k ) + a x ( k + 1 ) − x ( k ) T
v(k+1)=(1-a)v(k)+a\frac{x(k+1)-x(k)}{T}v(k+1)=(1−a)v(k)+aTx(k+1)−x(k)​

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