3.3.1) Z-transfer function of the I° Order Low Pass Digital Filter

OSpecpm fft Opm

 

(3.2.7.12)

0 1 10 ¹⁰

0 0.01 0.02 0.03

Output signal spectrum max OSpecpm







OSpecpm_k Specpm_k

fcpm

k fspm

N0gd

0 1 10 ¹⁰

4

2 0 2 4

Phase spectrum

0 arg OSpecpm



_k



arg Specpm



_k



fcpm

kfspm

N0gd

Fig.:3.2.7.5 Fig.:3.2.7.6

Wdbω3c^20 log Wlp j ω3c^

 



 

Wdbω3c 5.85 dB  ω_c is the angular frequency of the carrier

100 1 10 ³ 1 10 ⁴ 1 10 ⁵ 1 10 ⁶ 1 10 ⁷

40

20 0 20 40

Magnitude of W(ω)

20 log A3^

 

^3

0 20 log Wlp j ω^



(  )



Wlpp

ω31cfm ω3

ω

Fig.:3.2.7.7

3.3

Equivalent Digital Low Pass Filter (I°order)

3.3.1) Z-transfer function of the I° Order Low Pass Digital Filter

Chosen sampling period: T3s 1388.89 ns  place: Ts T3s Given the transfer function: Wlp s( ) A3 ω3

sω3

= , I can find its z-transform in this way:

with the change of variable: s 1 z ^1

= Ts ,we can place: 1 Ts

ωs 2 π

= , s ωs

2 π ^



1 z ^1



= ,

A3A3 Ts Ts ω3 ω3 s s zz Transfer function z-transform:

H1o z

 

^{A3 ω3}

s ω3

substitute s 1 z ^1

= Ts



collect z

A3 Ts ω3z z Ts ω3^



 1



^1





and after some algebraic manipulation and the definition of the following parameters:

α0 A3 ω3 Ts Ts ω3 1

 

 β0^



1 ω3 Ts 



^1 A310

α01.157482795 , β0 0.88425172 , you get the following result for the t. f. as a function of z:

H1o z

 

α0 1 1 β0 z  ^1



= Ts 1388.89 ns 

3.3 Equivalent Digital Low Pass Filter (I°order)

3.3.2) Difference equations (Low Pass filter(I°order)). Canonical form

Given the z transfer function H1o z( ) α0 1 1 β0 z  ^1



= , I can split it so that:

H1o z

 

^{Y z}

 

X z

 

= Y z

 

W_ z

 

^{W_ z}

 

X z

 

= 

and place: 1) Y z

 

W_ z

 

^{=α0 Y z}

 

=α0 W_ z

 

2) W_ z

 

X z

 

⁼



_{1 β0 z} ¹_ ^1



X z

 

=



1 β0 z  ^1



W_ z

 

=W_ z

 

β0 z ^1W_ z

 

which, inverting the z transform, gives :

x n

 

=w_ n

 

β0 w_ n 1







The corresponding set of difference equations is:

1) w n

 

=x n

 

^β0 w n 1^







2) y n

 

=α0 w n^

 

α0 α0 β0 β0

z ∞ α0 1

1 β0 z  ^1

 



 lim 

 α0

Fig.:3.3.2.1

DELPF1OCF

3.3 Equivalent Digital Low Pass Filter (I°order)

For each test signal there would be shown the following results:

1) Sequence of the periodic response,

2) Digital first order low pass filter difference equations, 3) Schematic,

4) Graphics,

5) Comparison of the Bode plots of the z and s transfer functions

3.3.3)

Sequence of the Voltage step response Digital first order low pass filter difference equations:

dimensionless input signal: u1_k^ Vstpsl n3



^kVpp



vi1 k

 

^u1k

 volt

10 0 10 20 30

0 2 4

Unit Pulses Sequence.

0 Vpp u1_k

0 8

k Vpp 4500 mV 

rows u1

 

256

fig.:3.3.3.1)

w1y1 DELPF1OCF vi1 A3 T3s

sec ω3 rad sec

 N0gd











w1w1y1 ⁰ y1 w1y1 ¹ α1



w1y1 ²



₀ β1



w1y1 ³



₀

0 25 50 75 100 0

10 20 30 40

Sequence of the function w

w1_k

k

0 25 50 75 100

50

40

30

20

10 0

Sequence of the response A3 Vi 0 y1_k

k

fig.:3.3.3.2 fig.:3.3.3.3

t1 0 τ3 0 τ3 100 τ3 10000



 100 τ3



0 4 10 ^5 8 10 ^5

0.06

0.04

0.02

Graph of the step response y(t)

time as multiple of τ

Output amplitude

A3 Vi 



e^11



 

 

A3 Vi ysr t1( )

yas t1( ) 0

τ3 5 τ3

t1

fig.:3.3.3.4

3.3 Equivalent Digital Low Pass Filter (I°order)

3.3.4) Sequence of the short Voltage pulse response.

TsTs3sp

t  τpw2  τpw2 4 τpw

 5000

 2 τpw



210^5 0 2 10 ^5 4 10 ^5 0

0.002 0.004 0.006

Sampled Input

Vi Vw t()

u44_k

τ3svp τpw τ3svp

t ns3sp _k fig.:3.3.4.1 Digital first order low pass filter difference equations:

dimensionless input signal: vi2 k

 

 u44_k

w2y2 DELPF1OCF vi2 A3 Ts3sp

 s ω3 s

rad

 N0gd







 

w2w2y2 ⁰ y2w2y2 ¹ α2



w2y2 ²



₀ β2



w2y2 ³



₀

Fig.:3.3.4.2

0 2 10 ^5 4 10 ^5 0

0.05 0.1 0.15

Sequence of w

w2_k

τ3svp τpw τ3svp

ns3sp_k

0 2 10 ^5 4 10 ^5

0.04

0.02 0

Sequence of the response

A3 Vi y2_k

τ3svp τpw τ3svp

ns3sp_k

fig.:3.3.4.3 fig.:3.3.4.4

t  τpw2  τpw2 4 τpw

 5000

 2 τpw



0 2 10 ^5 4 10 ^5

0.04

0.02 0

Graph of the Short Pulse Response

time as multiple of τ

Output amplitude

Vi

A3 Vi τpw τ3svp

τ3svp

Vi 0.01 V Vpp 4.5 V

fig.:3.3.4.5

3.3 Equivalent Digital Low Pass Filter (I°order)

3.3.5)

Sequence of the Bipolar Pulse train response TsTssqw Ttest 13.33 μs  Ts 138.89 ns  τ3 10.61 μs  fs 1

 Ts ωs 2 π fs Chosen test signal period, Ttest 13333.33 ns  1

Ttest 0.08 MHz Laplace transform of the test signal: Vip s

 

^Vi

s tanh Ttest s

4







 



u3k^ Vsqwb nsqw



_k



Pulse amplitude ±V_i:

0 100 200

510^3 0 5 10 ^3

Pulses Sequence.

0 Vi

u3_k

0

k fig.:3.3.5.1 Digital first order low pass filter difference equations:

dimensionless input signal: vi3 k

 

u3k

w3y3 DELPF1OCF vi3 A3 Tssqw

 s ω3 s

rad

 N0gd







 

w3 w3y3 ⁰ y3 w3y3 ¹ α3



w3y3 ²



₀ β3



w3y3 ³



₀

α30.12920836 , β3 0.987079164 , you get the following result for the t. f. as a function of z:

H1o z

 

α3 1 1 β3 z  ^1



 Ts 138.89 ns 

N0gd 256

Fig.:3.3.5.2

0 63.75 127.5 191.25 255

0.1

Sequence of the response

0

 ωtest 0.47 Mrads

 sec

Normalized Magnitude of H(z) and W(jω) 03

3.3 Equivalent Digital Low Pass Filter (I°order)

3.3.6)

Sequence of the Cusp test signal response. Definition:

u4_k vincsp_k

TsTocsp A310 t3  Ttest4  Ttest4 8 Ttest

 1000

 4 Ttest



0 1.333 10 2.667 10^5  ^54 10 ^55.333 10 ^5 0

2 4

Test signal

Vi fx1 t3( )

pwtcTtest

t3

0 1 10 ^52 10 ^53 10 ^54 10 ^5 0

2 4

Sampled test signal

Vi V u4_k

Ttest pwtc

nincsp_k

Fig.:3.3.6.1 Fig.:3.3.6.2

Cusps sequence of amplitude V_i:

Digital first order low pass filter recurrence relations:

dimensionless input signal: vi4 k

 

 u4_k fs

f3c 96 w4y4 DELPF1OCF vi4 A3 Tocsp

 s ω3 s

rad

 N0gd







 

w4 w4y4 ⁰ y4w4y4 ¹ α4



w4y4 ²



₀ β4



w4y4 ³



₀

α40.12920836 , β4 0.987079164 , you get the following result for the t. f. as a function of z:

H1o z

 

α4 1 1 β4 z  ^1



 Ts 138.89 ns 

Fig.:3.3.6.3

0 1 10 ^52 10 ^53 10 ^54 10 ^5 0

50 100 150

Sequence of the state function w

Fig.:3.3.6.4

0 1 10 ^52 10 ^53 10 ^54 10 ^5

20

15

10

5 0

Sequence of the periodic response A3 Vi y4_k

Vocsp nincsp



_k



nincsp_k Fig.:3.3.6.5

1.333 10 ^6 1.433 10 ^5 2.733 10 ^5 4.033 10 ^5 5.333 10 ^5

Normalized Magnitude of H(z) and W(jω) 03

3.3 Equivalent Digital Low Pass Filter (I°order)

3.3.7)

Sequence of the Sawtooth response Ts Tssw Sawtooth sequence of amplitude V_i:

Digital first order low pass filter recurrence relations:

fs f3c 96 dimensionless input signal: vi5 k

 

^u10k

 V

w5y5 DELPF1OCF vi5 A3 Tssw

 s ω3 s you get the following result for the t. f. as a function of z:

H1o z

 

α5 1

Sequence of the response

0 y5_k

voswc nsw

 

_k

nsw_k

Fig.:3.3.7.2 Fig.:3.3.7.3 Spec5o FFT y5

 

0 1 10 ⁶ 2 10 ⁶ 3 10 ⁶ 0

1 2

Amplitude Spectrum max Spec5o







Spec5o_k ftest

k fs

N0gd

0 1 10 ⁶ 2 10 ⁶ 3 10 ⁶

4

2 0 2 4

Phase spectrum

arg Spec5o



_k



0 ftest

k fs

N0gd

Fig.:3.3.7.4 Fig.:3.3.7.5

max Spec5o







max Spec5o







^{1 Ts 0.14 μs }

100 1 10 ³ 1 10 ⁴ 1 10 ⁵ 1 10 ⁶ 1 10 ⁷ 1 10 ⁸

40

20 0

Normalized Magnitude of H(z) and W(jω) 03



20 log H1o e



^{j ω Ts}



A3

















 20 log Wlp j ω( )

A3







 

20 log H1o e



^{j ωtest Ts}



A3



















ω3 ωtest

ω Fig.:3.3.7.6

3.3 Equivalent Digital Low Pass Filter (I°order)

3.3.8)

Sequence of the (single tone) Frequency Modulated carrier response mfm 8 dimensionless input signal: vi6 k

 

 u8k TsTsfm

w6y6 DELPF1OCF vi6 A3 Tsfm

 s ω3 s

rad

 N0gd







 

w6w6y6 ⁰ y6 w6y6 ¹ α6



w6y6 ²



₀ β6



w6y6 ³



₀

α60.03131754 , β6 0.996868246 ,

Digital first order low pass filter difference relations:

Fig.:3.3.8.1

you get the following result for the t. f. as a function of z:

H1o z

 

α6 1 1 β6 z  ^1



 Ts 33.33 ns 

0 4 10 ^6 8 10 ^6

0.2

0.1 0 0.1 0.2

FM signal

Vfm tfm

 

0 Tcfm

tfm 020 10 0

2 4 6 8 10

FM Spectrum (sinus. test signal)

A 4 A Jn ki mfm^







ki

Fig.:3.3.8.2 Fig.:3.3.8.3

X8 fft u8

 

0 100 200

0.2

0.1 0 0.1 0.2

Sampled FM signal

u8_k

k 0 5 10 ⁶ 1 10 ⁷ 1.5 10 ⁷

0 0.2 0.4

FM spectrum max X8( )

X8_k

fs f3c

k fsfm

N0gd

Fig.:3.3.8.4 Fig.:3.3.8.5

fsfm f3c 400

0 100 200

1

0.5 0 0.5

Sequence of the function w

0 100 200

0.02

0.01 0 0.01 0.02

Sequence of the response

Fig.:3.3.8.6 Fig.:3.3.8.7

f3c 0.08 MHz  fs

f3c 96 Spec6fft y6

 

mfm 8 ωfmm 942477.81

 s

0 2 10 ⁶ 4 10 ⁶ 6 10 ⁶ 0

0.01 0.02 0.03 0.04

FM Signal spectrum max Spec6( )

Spec6_k f3c

k fsfm

N0gd

0 5 10 ⁶ 1 10 ⁷ 1.5 10 ⁷

4

2 0 2 4

Phase spectrum

arg Spec6



_k



0 f3c

k fsfm

N0gd

Fig.:3.3.8.8 Fig.:3.3.8.9

max Spec6

 

0.09

100 1 10 ³ 1 10 ⁴ 1 10 ⁵ 1 10 ⁶ 1 10 ⁷ 1 10 ⁸

60

40

20 0

Normalized Magnitude of H(z) and W(jω) 03



20 log H1o e



^{j ω Ts}



A3

















 20 log Wlp j ω( )

A3







 

20 log H1o e



^{j ωtest Ts}



A3



















ω3 ωtest

ω Fig.:3.3.8.10

3.3 Equivalent Digital Low Pass Filter (I°order)

3.3.9)

Sequence of the Phase Modulated carrier response. mpm 6 dimensionless input signal: vi7 k

 

 u9k TsTspm

w7y7 DELPF1OCF vi7 A3 Tspm

 s ω3 s

you get the following result for the t. f. as a function of z:

H1o z

 

α7 1 1 β7 z  ^1



 Ts 0.02 ns 

107.4999999964.9999999922.4999999871.66799996 10 ^8

20

PM Spectrum (sinus. test signal)

A 4 A Jn ki mpm^







ki

mpm 6 Fig.:3.3.9.2 Fig.:3.3.9.3

0 100 200

Sequence of the state function w

w7_k

Sequence of the response

y7_k

3.3 Equivalent Digital Low Pass Filter (I°order)

3.3.10)

BODE plot (Low Pass Analog v. s. Digital filter(I°order)) 20 log α7

 

96.08 Ts 0.02 ns 

Normalized Magnitude of H(z) and W(jω) 03

The Phase of H(z)and W(jω)

π

For each test signal would be shown the following results:

1) Sequence of the periodic response,

2) Digital first order low pass filter difference equations, 3) Schematic,

4) Graphics,

5) Comparison of the Bode plots of the z and s transfer functions

α0 and β0 are functions of the sampling period which in turn it depends from the kind of signal used.

TsT3s

Numerator degree Nn 1 Denominator degree Md 1

N1Nn Md N0gd 256 h1k0

A generic first order transfer function in the z domain takes this form:

H z

 

^{b0b1z1}

a₀a₁z^1

=

The coefficients of the numerator and denominator can be defined as the elements of two vectors, namely a and b, hence:

Numerator coeffs Denominator coeffs n11 N0gd 1  b_n1=0.0 a_n1=0.0

b₀=α0 a₀=1

b₁=0 a₁=β0

and divide the two polynomial by means of the following algorithm:

N12 h10 b₀

y8ν 0 ν

k

if ν k



 0h1ku1νk0



 







IACAN

tfs1 IACAN u1

VA3T3sω3N0gd

 



α0



tfs1 ⁰



₀ β0



tfs1 ⁰



₁ S1



tfs1 ⁰



₂ E11



tfs1 ⁰



₃ h11 tfs1 ¹

a



tfs1 ²



b



tfs1 ³



T. F. Numerator coefficients:

a^T 1 -0.88 0 0 0 0 0 0 ...

T. F. Denominator coefficients:

b^{T 0 1 2 3 4 5 6 7 8}

0 -1.16 0 0 0 0 0 0 0 ...



Sequence of the Impulse response:

h1^T 0 0 0 0 0 ...

Stability (S1<∞): S1

0 rows h1( ) 1

k

h1k





= S110

Energy of the sequence h1: E11

0 rows h1( ) 1

k

h1k

 

²





= E116.14

0 5 10 ^5 1 10 ^4 1.037

 10⁶

7.5410⁵ 4.712

 10⁵ 1.885

 10⁵

9.425 10 ⁴ Impulse Response

A3 ω3 e

A3 ω3 w t1( )

A3ω3^



t1 ω3 1



0

τ3 5τ3

t1

0 10 20 30 40 50

1.5

1

0.5 0

Impulse Response Sequence

h11_k

k

3.4 Transfer Function Sequence obtained by an Algorithm. Convoluting Output.

In document 3 Equivalent Digital Low Pass Filter (I order) (Page 30-41)