IMPLEMENTATION STRUCTURES FOR DISCRETE-TIME SYSTEMS

IMPLEMENTATION STRUCTURES FOR DISCRETE-TIME SYSTEMS

"REAL-TIME IMPLEMENTATIONS"

"FLOW GRAPH PROPERTIES"

PARALLEL NETWORKS & CASCADE NETWORKS

IS COMPARABLE TO:

FEEDBACK NETWORKS

W(z) = X(z) E(z) + Y(z) G(z)

Y(z) =W(z) F(z) = (X(z) E(z) + Y(z) G(z)) F(z)

Y(z) - Y(z) G(z) F(z) = X(z) E(z) F(z)

Y(z) (1 - G(z) F(z)) = X(z) E(z) F(z)

COMBINED IDEAS

POSITIVE FEEDBACK LOOP:

E(z) = 1; F(z) = 1; G(z) = H₂(z)H₃(z)

PARALLEL/CASCADE COMBINATION

Y(z) = W(z)(H₁(z) + H₂(z)H₄(z))

TRANSPOSED FORMS

1. Rename inputs and outputs.

2. Redraw arrows

3. Rearrange in standard left-right format

H(z) =Y(z)/X(z) = d +z^-1 ct (I - z^-1A )^-1 b

Transpose Rules:

(A + B)^t = A^t + B^t

(AB) ^t= B^t A^t

(A^-1)^t =(A^t )^-1

H^t (z) = Y(z)/X(z)= d + b^tz^-1(I - z^-1A^t )^-1 c

We want to reach the stage where the following kind of thinking becomes "automatic"

THEN:...y(n) = 2 y(n-1) - 6 y(n-2) + 3 x(n)

THEN:...y(n) = 3 x(n) - 4 x(n-1) + 5x(n-2)

THEN:...y(n) = 2 y(n-1) - 6 y(n-2) + 3 x(n) - 4 x(n-1) + 5x(n-2)

One implementation for:...y(n) = 2 y(n-1) - 6 y(n-2) + 3 x(n)

THEN:...y(n) = 2 y(n-1) - 6 y(n-2) + 3 x(n) - 4 x(n-1) + 5x(n-2)

IF: H(z) = H₁(z)H₂(z)

THEN:...v(n) = 3 x(n) - 4 x(n-1) + 5x(n-2)

THEN:...y(n) = 2 y(n-1) - 6 y(n-2) + v(n)

IF:...H(z) = H₂(z)H₁(z)

THEN:...w(n) = 2 w(n-1) - 6 w(n-2) + x(n)

THEN:...y(n) = 3 w(n) - 4 w(n-1) + 5w(n-2)

THEN:

y(n) = -a₁ y(n-1) + -a₂ y(n-2) + b₀ x(n) + b₁ x(n-1) + b₂x(n-2)

OR: DIRECT FORM II:

COMPARED TO DIRECT FORM I = TRANSPOSE OF DIRECT FORM II

CASCADE FORMS FOR REAL POLYNOMIALS

M = M₁ + 2M₂ and N = N₁ + 2N₂

And if N >= M:

= b0

THEN:

y₁(n) = -a₁₁ y₁(n-1) - a₂₁ y₁(n-2) + b₀ x(n) + b₁₁ x(n-1) + b₂₁ x(n-2)

y₂(n) = -a₁₂ y₂(n-1) - a₂₂ y₂(n-2) + y₁(n) + b₁₂ y₁(n-1) + b₂2 y₁(n-2)

y₃(n) = -a₁₃ y₃(n-1) - a₂₃ y₃(n-2) + y₂(n) + b₁₃ y₂(n-1) + b₂₃ y₂(n-2)

An array of the number of storage nodes might look like the following;

k=1 k=2 k=3 .

in x (n-2) x (n-1) x (n)

out in y₁(n-2) y₁(n-1) y₁(n)

out in y₂(n-2) y₂(n-1) y₂(n)

out y₃(n-2) y₃(n-1) y₃(n)

MATLAB IMPLEMENTATION OF CASCADE FORM

%% clfild97.m file

% SINUSOIDAL INPUT

F = [ 0 0 0; 0 0 0; 0 0 0; 0 0 0];

for n=0:30

x=sample(n);

[y,F] =clfilt97(x,F); % PLUS PLOT STUFF

end;

% IMPULSE INPUT

F = [ 0 0 0; 0 0 0; 0 0 0; 0 0 0];

for n=0:30

x=0;

if n==2, x=2; end;

[y,F] =clfilt97(x,F) %PLUS PLOT STUFF

end;

function y=sample(n);

y=cos(pi/8*n) + cos(6*pi/8*n);

function [y,F] = clfilt97(x,F);

b0=[.0904 .0904 .0904]; b1= 2*b0; b2=b0;

a1=[ -1.268 -1.010 -.9044]; a2= [.7051 .3585 .2155];

F(1,3) = x;

for k=1:3

in=k; out=k+1;

F(out,3)=b0(k)*F(in,3)+b1(k)*F(in,2)+b2(k)*F(in,1)...

- a1(k)*F(out,2) - a2(k)*F(out,1);

end;

y=F(4,3)

pause(1);

for k=1:4

F(k,1)=F(k,2);

F(k,2)=F(k,3);

end;

PARALLEL FORMS/MORE WORK---PARTIAL FRACTIONS

And there is nothing to keep us from using the following form, grouping real zeros and poles together, and considering zero value coefficients and assuming that M=N:

THEN:	y₁(n) = a₁₁ y₁(n-1) + a₂₁ y₁(n-2) + e₀₁ x(n) + e₁₁ x(n-1)

	y₂(n) = a₁₂ y₂(n-1) + a₂₂ y₂(n-2) + e₀₂ y1(n) + e₁₂ y₁(n-1)

	y₃(n) = y₁(n) + y₂(n) + e₀ x(n)

PROPERTIES OF NETWORK COEFFICIENTS

All factors can have form:

F_i(z) = 1 + c_1i z^-1 + c_2i z^-2 = (1- q_1i z-1 )( 1 - q_2i z-1)

c_1i = -(q_1i + q_2i ) and c_2i = q_1i q_2i

if c_1i ² > 4 c_2i roots are real and

if c_1i ² < 4 c_2i roots are complex conjugatves: q_1i = q_2i* = q_i

c_1i = - 2 Re ( q_i) = -2 r_i cos (f_i) and c_2i = |q_i| ² = r_i²

FOR DENOMINATOR FACTORS FOR STABILITY POLES MUST BE INSIDE THE UNIT CIRCLE:

D_i(z) = 1 + a_1i z^-1 + a_2i z^-2 = (1- p_1i z^-1 )( 1 - p_2i z^-1)

|p_1i z^-1 | and | p_2i z^-1| < 1 or |a_2i | = |p_1i p_2i | <1

ALSO CAN BE SHOWN USING:

THAT |a_1i | < 1 + a_2i

ZEROS TEND TO BE ON THE UNIT CIRCLE

N_i(z) = 1 + b_1i z^-1 + b_2i z^-2 = (1- z_1i z^-1 )( 1 - z_2i z^-1)

= 1 - 2 cos f_i z^-1 + z^-2

= 1 ± 2 z^-1 + z^-2 if roots are real and equal.

= 1 - z^-2 if roots are real and unequal (-1, +1).

= 1 ± z^-1 if roots are real and first order.

WHERE ARE ROOTS LOCATED IN THE FOLLOWING?

F_i(z) = 1 + 1.414 z^-1 + z^-2

F_i(z) = 1 + 2 z^-1 + z^-2

F_i(z) = 1 -.8 z^-1 + .64 z^-2

F_i(z) = 1 -1.08333 z^-1 + 0.25 z^-2

BASIC NETWORKS FOR FIR SYSTEMS - LINEAR PHASE

h(n)	= {b_n for n= 0, 1, ....,M
	= {0 otherwise

FIRS and LINEAR PHASE DELAY --- THE REAL ADVANTAGE

Examples: b_m = ± b*_M-m for n=0,1,M

IF M IS EVEN and SYMMETRY IS EVEN = TYPE I

h(n) = d(n) + 2d(n-1) + 4d(n-2) + 2d(n-3) + d(n-4)

IF M IS ODD and SYMMETRY IS EVEN = TYPE II

h(n) = d(n) + 2d(n-1) + 4d(n-2) + 4d(n-3) + 2d(n-4)+ d(n-5)

IF M IS EVEN and SYMMETRY IS ODD = TYPE III

h(n) = d(n) + 2d(n-1) + 0 d(n-2) - 2d(n-3) - d(n-4)

IF M IS ODD and SYMMETRY IS ODD = TYPE IV

h(n) = d(n) + 2d(n-1) + 4d(n-2) - 4d(n-3) - 2d(n-4)- d(n-5)

ASSOCIATED FREQUENCY RESPONSES

NOTE LINEAR PHASES AND 180^o JUMPS

NOTE CONSTRAINTS AT z = 1 and z = -1 (w = 0 and w = p)

ALWAYS:

so for odd symmetry cases (III and IV)

Always:

so for types II and III: H(-1) = 0

b_m = ± b*_M-m is the same as: h(n) = ± h*(M-m)

or H(z) = ± z ^-M H*(1/z*) or z ^MH(z) = ± H*(1/z*)

WHICH SAYS THAT IF z is a zero, then 1/z* is also a zero of the function.

TYPE I EXAMPLE WITH REAL NUMBERS

H(z) = 1 + 2z^-1 + 4z^-2 + 2z^-3 + z^-4

= z^-2 (z² + 2z¹ + 4 + 2z^-1 + z^-2)

= z^-2 (z² + z^-2 + 4 + 2z¹ + 2z^-1)

= z^-2 [(z² + z^-2)+ 4 + (2z¹ + 2z^-1)]

or H(ejw) = e^-2j^w [(e^2j^w + e^-2j^w)+ 4 + (2^ej^w + 2e^-j^w) )

or = e^-2j^w [2cos2w + 4 + 4cosw ) = e^-2j^w R(w)

< H(ejw) = -2w + C = -Mw/2 + C

where C = 0 or ¹ depending on sign of R

which is linear with 180 degree jumps.

We are really trying to constrain what is defined as a filter's group delay:

NOTE CHANGES IF bs are not real:

TYPE II EXAMPLE

h(n) = d(n) + 2d(n-1) + 4d[n-2) + 4d(n-3)+ 2d(n-4) + d(n-5)

H(z) = 1 + 2z^-1 + 4z^-2 + 4z^-3 + 2z^-4 + z^-5

= z^-5/2 (z^5/2 + 2z^3/2 + 4z^1/2 + 4z^-1/2 + 2z^-3/2 + z^-5/2)

=z^-5/2 (z^5/2 + z^-5/2+ 2z^3/2 + 2z^-3/2 + 4z^1/2 + 4z^-1/2 )

H'(w) =e^-5j^w/2[(e ^5j^w/2+e ^-5j^w/2)+2(e ^3j^w/2+e^-3j^w/2)+4(e^j^w/2+e^-j^w/2))

= e^-5j^w/2 [2cos5w/2 + 4cos3w/2 + 8cos w/2 ]

TYPE III EXAMPLE WITH REAL NUMBERS

H(z) = 1 + 2z^-1 - 2z^-3 - z^-4

= z^-2 (z² + 2z¹ - 2z^-1 - z^-2)

= z^-2 (z² - z^-2 + 2z¹ - 2z^-1)

= z^-2 [(z² - z^-2)+ (2z¹ - 2z^-1) )

or H(ejw) = e^-2j^w [(e^2j^w - e^-2j^w)+ (2e^j^w - 2e^-j^w) ]

or = e^-2j^w [j2sin2w + j4sin w ] = j e^-2j^w R(w)

TYPE IIIs are used for differentiators:

NOTE: IDEAL DIFFERENTIATIOR: H'_D(w) = jw

CHECK FIRST DIFFERENCE:

y(n) = x(n) - x(n-1)

h(n) = d(n) - d(n-1)

H(z) = 1 - z^-1

H(e^j^w) = 1 - e^-j^w

H(e^j^w) = e^-j^w/2 (e ^j^w/2 - e^-j^w/2)

= 2 e^-j^w/2 (jsin w/2) = j(group delay) R(w)

R(w) = 2 sin w/2 which is approximately w for small w.

GENERAL EQUATIONS:

where h(k) = h(M-k)

and notation for corresponding x(n-k) is x(n-M+k)

For even M:

For odd M:

SPECIAL FILTERS

ALLPASS FILTERS: |H(e^j^w)| = |H'(w)| = 1 for all w

If zs is a zero of H(z) then 1/zs is a pole of H(z)

Example:

COMB FILTER -- EVENLY SPACED NOTCHES

Select H(z) and then substitute to get H(z^k)

which has a frequency response H(e^j^wk) = H'(kw)

Since H'(w) has a period of 2¹, H'(kw) has a period of 2p/k.

Example of a simple highpass filter:

has zeros at k roots of 1 and poles at the k roots of a.

Example for k=6 and a=.3

(POWER) COMPLEMENTARY FILTERS

|H'₁(w)|² + |H'₂(w)|² = 1 for all w, equivalently

H₁(z) H₁(z^-1) + H₂(z) H₂(z^-1) = 1

Example where: H₁(z) is lowpass and H₂(z) is highpass.

CRITERIA: H₁(z) and H₂(z) must be bounded real, ie

|H'_i(w)| <= 1

H₁(z) H₁(z-1) + H₂(z) H₂(z-1) = 1

or: P(z)P(z^-1)+ Q(z)Q(z^-1) = D(z)D(z^-1)

or: Q(z)Q(z^-1) = D(z)D(z^-1) - P(z)P(z^-1)

As many classical filters have all their zeros on the unit circle,
Q(z)Q(z^-1) will have double zeros on the unit circle.

CAN BE SHOWN: Hi(1) = 1 and Hi(-1) = 0 , then there exists allpass filters such that:

H₁(z) = [ A₁(z) + A₂(z) ]/2

H₂(z) = [ A₁(z) - A₂(z) ]/2

A₁(z) = H₁(z) + H₂(z)

A₂(z) = H₁(z) - H₂(z)

First Order Example:

H1(1) = 1 and H1(-1) = 0

H2(1) = 0 and H2(-1) = 1

P(z)+Q(z) = 1 - az-1 = D(z)

P(z) - Q(z) = z-1 -a

Therefore:

y₁(n)	= -a₁₁ y₁(n-1) - a₂₁ y₁(n-2) + b₀ x(n) + b₁₁ x(n-1) + b₂₁ x(n-2)

y₂(n)	= -a₁₂ y₂(n-1) - a₂₂ y₂(n-2) + y₁(n) + b₁₂ y₁(n-1) + b₂2 y₁(n-2)

y₃(n)	= -a₁₃ y₃(n-1) - a₂₃ y₃(n-2) + y₂(n) + b₁₃ y₂(n-1) + b₂₃ y₂(n-2)

k=1	k=2	k=3 .
in			x (n-2)	x (n-1)	x (n)
out	in		y₁(n-2)	y₁(n-1)	y₁(n)
	out	in	y₂(n-2)	y₂(n-1)	y₂(n)
		out	y₃(n-2)	y₃(n-1)	y₃(n)

N_i(z)	= 1 + b_1i z^-1 + b_2i z^-2 = (1- z_1i z^-1 )( 1 - z_2i z^-1)

	= 1 - 2 cos f_i z^-1 + z^-2

	= 1 ± 2 z^-1 + z^-2 if roots are real and equal.

	= 1 - z^-2 if roots are real and unequal (-1, +1).

	= 1 ± z^-1 if roots are real and first order.

H(z)	= 1 + 2z^-1 + 4z^-2 + 2z^-3 + z^-4

	= z^-2 (z² + 2z¹ + 4 + 2z^-1 + z^-2)

	= z^-2 (z² + z^-2 + 4 + 2z¹ + 2z^-1)

	= z^-2 [(z² + z^-2)+ 4 + (2z¹ + 2z^-1)]

or H(ejw)	= e^-2j^w [(e^2j^w + e^-2j^w)+ 4 + (2^ej^w + 2e^-j^w) )

or	= e^-2j^w [2cos2w + 4 + 4cosw ) = e^-2j^w R(w)

< H(ejw)	= -2w + C = -Mw/2 + C

h(n)	= d(n) + 2d(n-1) + 4d[n-2) + 4d(n-3)+ 2d(n-4) + d(n-5)

H(z)	= 1 + 2z^-1 + 4z^-2 + 4z^-3 + 2z^-4 + z^-5

	= z^-5/2 (z^5/2 + 2z^3/2 + 4z^1/2 + 4z^-1/2 + 2z^-3/2 + z^-5/2)

	=z^-5/2 (z^5/2 + z^-5/2+ 2z^3/2 + 2z^-3/2 + 4z^1/2 + 4z^-1/2 )

H'(w)	=e^-5j^w/2[(e ^5j^w/2+e ^-5j^w/2)+2(e ^3j^w/2+e^-3j^w/2)+4(e^j^w/2+e^-j^w/2))
	= e^-5j^w/2 [2cos5w/2 + 4cos3w/2 + 8cos w/2 ]

H(z)	= 1 + 2z^-1 - 2z^-3 - z^-4

	= z^-2 (z² + 2z¹ - 2z^-1 - z^-2)

	= z^-2 (z² - z^-2 + 2z¹ - 2z^-1)

	= z^-2 [(z² - z^-2)+ (2z¹ - 2z^-1) )

or H(ejw)	= e^-2j^w [(e^2j^w - e^-2j^w)+ (2e^j^w - 2e^-j^w) ]

or	= e^-2j^w [j2sin2w + j4sin w ] = j e^-2j^w R(w)

y(n)	= x(n) - x(n-1)

h(n)	= d(n) - d(n-1)

H(z)	= 1 - z^-1

H(e^j^w)	= 1 - e^-j^w

H(e^j^w)	= e^-j^w/2 (e ^j^w/2 - e^-j^w/2)
	= 2 e^-j^w/2 (jsin w/2) = j(group delay) R(w)

R(w)	= 2 sin w/2 which is approximately w for small w.