Question 4 of 6: Stability and Structures for a Parallel System

Question 4: Stability and Structures for a Parallel System (12 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A causal system already written as the sum of a first-order accumulator and a second-order section.

Find. A justified stability verdict, a parallel-form signal flow graph with a direct-form-II biquad, and a cascade-form signal flow graph with a transposed direct-form-II biquad.

Approach. Locate every pole to settle stability, then realise the given sum directly for the parallel form and combine the two sections over a common denominator for the cascade form.

1. Part (a): locate the poles. The second-order denominator gives $$z^{2}-z+0.8=0\ \Rightarrow\ z=\frac{1\pm\sqrt{1-3.2}}{2}=0.5\pm j0.7416,\qquad |z|=\sqrt{0.8}=0.8944 ,$$ which is safely inside the unit circle. The first-order section, however, contributes a pole at $z=1$, exactly on the unit circle.
2. Draw the stability conclusion. For a causal system the region of convergence is the exterior of the outermost pole, here $|z| \gt 1$, which does not include the unit circle. Equivalently, $1/(1-z^{-1})$ inverts to the unit step $u[n]$, and $\sum_n|u[n]|$ diverges, so $$\boxed{\text{the system is NOT stable: the accumulator pole at } z=1 \text{ lies on the unit circle}}$$ A bounded input such as a unit step produces an output that grows without bound, which is the physical statement of the same fact.
3. Part (b): write the two difference equations. The parallel form realises the given sum as drawn, one branch per term. The accumulator is $$y_1[n]=x[n]+y_1[n-1],$$ and the biquad in direct form II uses the shared state $w[n]$: $$w[n]=x[n]+w[n-1]-0.8\,w[n-2],\qquad y_2[n]=w[n]-w[n-1],$$ with the output being $y[n]=y_1[n]+y_2[n]$. Direct form II is the canonic realisation: the poles are implemented first and the zeros are then tapped off the same delay chain, so only two delays are needed for the second-order section.
4. Draw the parallel-form flow graph. The input feeds both branches and their outputs are summed.
   Figure 4.2 — parallel form: a first-order all-pole section (top) and a direct-form-II second-order section (bottom), summed at the output. Three delays in total.
5. Part (c): combine the two sections over a common denominator. Adding the fractions, $$H(z)=\frac{\left(1-z^{-1}+0.8z^{-2}\right)+\left(1-z^{-1}\right)^{2}}{\left(1-z^{-1}\right)\left(1-z^{-1}+0.8z^{-2}\right)}=\frac{2-3z^{-1}+1.8z^{-2}}{\left(1-z^{-1}\right)\left(1-z^{-1}+0.8z^{-2}\right)} .$$ The numerator has complex roots (its discriminant is $9-4(2)(1.8)=-5.4$), so it cannot be split into two real first-order factors. The natural cascade therefore puts the whole numerator on the second-order section: $$H(z)=\underbrace{\frac{1}{1-z^{-1}}}_{\text{1st order}}\cdot\underbrace{\frac{2-3z^{-1}+1.8z^{-2}}{1-z^{-1}+0.8z^{-2}}}_{\text{2nd order}} .$$ The zeros lie at $z=0.75\pm j0.5809$, of magnitude $0.9487$.
6. Write the transposed direct-form-II equations. With $v[n]$ the output of the first section, $v[n]=x[n]+v[n-1]$, the transposed structure computes the output first and pushes the partial sums down the delay chain: $$y[n]=2v[n]+s_1[n-1],\quad s_1[n]=-3v[n]+y[n]+s_2[n-1],\quad s_2[n]=1.8\,v[n]-0.8\,y[n].$$ Transposition reverses every branch and swaps the roles of input and output; it preserves the transfer function exactly while changing the internal node scaling, which is why it is often preferred in fixed-point implementations.
7. Draw the cascade-form flow graph. Driving both structures with an impulse and comparing the first forty samples confirms that the parallel and cascade realisations produce the identical impulse response, and that this response does not decay — the numerical signature of the unstable pole found in part (a).
   Figure 4.3 — cascade form: the first-order all-pole section drives a transposed direct-form-II second-order section carrying the combined numerator.