Digital Averaging Filter

Contents Background Design& Implementation Refine Discussion

What’s RMS (Root Mean Square) RMS is the constant value that yields the same power dissipation as the timeaveraged power dissipation of the signal For continuous signal

f rms

2 T2 1 f (t ) dt T2 T1 T1

For 1 cycle sinusoidal signal

RMS

f0

T1 1 f 0

T1

A cos ( 2 π f 0 t ) dt

2

A 2

What’s Average Average is the DC component of the signal For continuous signal

f avg

T2 1 f (t )dt T2 T1 T1

For 1 cycle sinusoidal signal (abs)

AVG f 0

T1 1 f 0

T1

A cos ( 2 π f 0 t ) dt

2 A

RMS vs. AVG For 1 cycle sinusoidal signal (abs)

RMS

A 2

AVG

2 A

If we know AVG, we know RMS

RMS

2 2

AVG 1.11072 AVG

Get Average in Analog World To get abs -> Rectifier

To get DC component -> Low Pass Filter

Get Average in Digital World To get abs -> Reverse the sign of negative sample

To get DC component -> Digital Low Pass Filter

Reverse Sign Reverse the sign of negative sample DC component is average value Cons - Introduce harmonics - Low precision in case of high harmonics Pros - No impact by a variety of frequency - No impact by a variety of initial phase angle MATLAB Example - a_ReverseSign.m

Design LPF– Overview Find a continue-time filter model, create transfer function Convert continue-time transfer function to discrete-time transfer function Write difference equation from discrete-time transfer function Difference equation can be implemented by program.

Design LPF– Continuous-Time Model Design a simple analog filter

Find transform function in s domain (Laplace Transform)

1 1 c H a ( s) s C R C 1 1 s c R s s C R C

Design LPF– Discrete-Time Model Convert Ha(s) to H(z) The convert method - Impulse invariance (DC gain is not 1)

A H a ( s) k k 1 s skH a ( s)

N

H ( z)

Ts Ak sk Ts z 1 k 1 1 eN

c s c

H ( z)

1 e c Ts z 1

c Ts

- Bilinear transform (DC gain is 1)

2 1 z 1 s Ts 1 z 1H a ( s)

2 1 z 1 H ( z) H a ( ) Ts 1 z 1

c s c

c Ts c Ts z 1 p p z 1 H ( z) 2 c Ts (2 c Ts ) z 1 1 (1 2 p) z 1

p

c Ts 2 c Ts

Design LPF– Difference Equation From H(z) to Difference EquationH ( z) b(1) b(2) z 1 ... b(nb 1) z nb X ( z ) Y ( z) 1 a(2) z 1 ... a(na 1) z na

Y (n) b(1) * X (n) b(2) * X (n 1) ... b(nb 1) * X (n nb ) a(2) *Y (n 1) ... a(na 1) *

Y (n na )

Design LPF– Difference Equation (cont’d) Impulse invarianceH ( z) 1 e c Ts z 1

c Ts

Y (n) c Ts X (n) e c Ts Y (n 1)

c Ts c Ts z 1 p p z 1 H ( z) 2 c Ts (2 c Ts ) z 1 1 (1 2 p) z 1 Bilinear transform

p

c Ts 2 c Ts

Y (n) p X (n) p X (n 1) (1 2 p) Y (n 1)

p

c Ts 2 c Ts

Implement LPF (Impulse invariance) MATLAB Example - b_ImpulseInvariance.m - Multiplication X 2 - Add X 1 Problems - The result is not stable (Filter is not ideal) - Multiplication operation costs too much CPU time

Design stable LPF Decrease cutoff frequency is not a good idea due to the long stable time Use cascading filters n Order -> -dB X n Y1(n) c Ts X (n) e c Ts Y1(n 1)

Y 2(n) c Ts Y1(n) e c Ts Y 2(n 1)

Y 3(n) c Ts Y 2(n) e c Ts Y 3(n 1) MATLAB Example - c_CascadingFilters.m

Design LPF without Multiplication Zero-Init Response& Zero-State Response

kTs kTs - Ts

vzi (t ) vo e

t R Ct t

1 vzs (t ) h(t ) (vi (t )) vi h( ) d vi e R C d vi (1 e R C ) 0 R CTs R C

t

vzi (kTs ) vo (kTs Ts ) e

vzs (kTs ) vi (kTs Ts ) (1 e

Ts R C

)

Design LPF without Multiplication (cont’d)●Write difference equation from response equation

vo (kTs ) vzi (kTs ) vzs (kTs ) vo (kTs Ts ) e

Ts R C

vi (kTs Ts ) (1 e

Ts R C

)

vo (k ) vo (k 1) e

Ts R C

vi (k 1) (1 e

Ts R C

)Exponential smoothing/wiki/Exponential_smoothing

Y (n) X (n 1) (1 ) Y (n 1) 1 e Ts c

Design LPF without Multiplication (cont’d)●Multiplication is replaced by bit-shift - Bit-Shift X 1 - Add X 2

Y (n) (( X (n 1) Y (n 1)) p) Y (n 1) 2 p●Cutoff frequency

c Fs ln(1 2 p ) fc 6.2414 T 5000, p 7 2 2 s

●MATLAB Example - d_ExponentiallyWeightedMovingAverage.m

Noise DC offset - Can be removed with an addition cascading filters Normal Distributed Noise - Some impact to average value Harmonics - Huge impact to average value MATLAB Example - e_ImpactFromNoise

Other Ways to Design Filter MATLAB Transfer Function Example - f_MatlabFilter.m MATLAB Filter Function butter -> Butterworth filter design cheby1 -> Chebyshev Type I filter design (pass band ripple) Window design method ( Finite Impulse Response )