Is this system causal or not?

Question

My efforts of solving this question are below.

I came to a conclusion that this system is causal, since:

$$ \begin{cases} w[k]+5w[k-1]+6w[k-2]=x[k] \\ y[k]=w[k]+2w[k-1]+3w[k-2]+4w[k-3] \end{cases} $$

Where $w[n]$ is the output of the leftmost summation. After conducting Z transforms:

$$ \begin{cases} \frac{W(z)}{X(z)} = \frac{1}{1+5z^{-1}+6z^{-2}} \\ \frac{Y(z)}{W(z)} = 1+2z^{-1}+3z^{-2}+4z^{-3} \end{cases} $$

$$\Rightarrow H(z) = \frac{Y(z)}{X(z)} = \frac{Y(z)}{W(z)} \frac{W(z)}{X(z)} = \frac{1+2z^{-1}+3z^{-2}+4z^{-3}}{1+5z^{-1}+6z^{-2}} = \frac{z^{3}+2z^{2}+3z+4}{z^{3}+5z^{2}+6z} = \frac{(z+1.651)(z+(0.175-j1.547))(z+(0.175+j1.547))}{z(z+2)(z+3)} $$

According to that, I had concluded that this system has 3 zeroes and 3 poles, therefore the number of zeroes is less than or equal to the number of poles, and therefore the system is causal.

However, the answer key states the following:

Where have I made a mistake? Please, correct me if I'm wrong.

Answer 1

Note that in this case you can see that the system is causal only from the given implementation. It's important to understand that you can't see it from the difference equation (if no initial conditions are given), and in general you can't see it from the transfer function either (if no region of convergence is given). The only case for which the expression of the transfer function is sufficient to conclude that the corresponding system is non-causal is if the transfer function is not proper, i.e., if the degree of the numerator (as a function of $z$, not $z^{-1}$!) is greater than the degree of the denominator. Also take a look at this related answer.

Let's look at an example:

$$y[n]=ay[n-1]+bx[n]\tag{1}$$

Does $(1)$ describe a causal system? Well, it can describe a causal system, but it can also be read as

$$y[n-1]=\frac{1}{a}\big(y[n]-bx[n]\big),\qquad a\neq 0\tag{2}$$

which can clearly describe a non-causal system if interpreted as "at time instant $n-1$, compute $y[n-1]$ from $y[n]$ and $x[n]$ according to the equation".

The corresponding transfer function

$$H(z)=\frac{b}{1-az^{-1}}\tag{3}$$

just conveys the same information as the difference equation $(1)$. Consequently, the expression $(3)$ doesn't reveal the system's causality either. Only if we know the region of convergence (ROC) of $(3)$ can we judge whether the system is causal or not. If the ROC is $|z|>a$, the corresponding system is causal, and and if the ROC is $|z|<a$, the system is non-causal.

Answer 2

Purely by inspection of the block diagram the system is causal, because the output is the sum of the current input sample and stuff that's delayed -- there's no $z$ blocks in there to predict the future, just $z^{-1}$ block to react to the past.

Also by your method of finding the transfer function, the system is causal -- with a $3^{rd}$ order numerator and a $3^{rd}$ order denominator, it does have an immediate response, but it's not "reacting" to future events.

I believe that whoever made the exam key erred by calculating polynomials in $z^{-1}$, in which case the denominator order is, indeed, smaller than the numerator order, but then applying a rule that's only valid if the polynomials are in $z$.

(Note: some authors may want to define a strictly causal system as one whose output only depends on past values of the input -- that doesn't appear to be the case here; your exam key seems to be going by the correct rule, just misusing it).

Answer 3

The system is causal, provided that the recursion is forward; i.e., it's recursed for increasing $k$.

Seeing that you are confused about causality tests, let me elaborate on it.

Let's put the definition of causality from Oppenheim's Signals & Systems book :

A system is causal if the output at any time depends only on values of the input at the present time or past

Let's also put the definition of causality from Oppenheim's Discrete-Time Signal Processing book :

A system is causal if, for every choice of $n_0$, the output sequence value at the index $n=n_0$ depends only on the input sequence values for $n \leq n_0$.

Focusing on discrete-time systems, then we shall consider linear, nonlinear, time-varying, time-invariant, recursive, non-recursive systems while testing for causailty.

In general if nothing about the system is specified but just a formula for input-output computation (I/O relation) is given, then you can use the above definitions to test whether the system is causal or not.

For example the system : $$ y[n] = T\{x[n]\} = \sum_{k=n-d}^{n+d} c[k]~x[k] $$ is a non-recursive, linear, and non-causal (and stable) system with memory, as the summation requires future values of input.

For example the system : $$ y[n] = T\{x[n]\} = \sum_{k=-d}^{d} x[n+k]x[n-k] $$ is a non-recursive, non-linear, and non-causal (and stable) system with memory, as the summation requires future values of input.

Note that for a non-recursive system, the output $y[n]$ only depends on the values of input $x[n]$, and not on itself: Such systems do not require an initial condition (of $y[n]$) to be specified for computation of the output. However, for recursive systems you also need to specify initial conditions of $y[n]$, for computing the output via a recursion.

In particular if a system is LTI (Linear Time-Invariant) you can use it's impulse response $h[n]$ to test whether it's causal or not; specifically:

An LTI system is causal iff $h[n]=0$ for all $n<0$

Furthermore, for LTI systems, you can compute their system function $H(z)$, (Z-transform of impulse response) and test for causality as:

An LTI system with rational transfer function is causal if the ROC (region of convergence) of its system function H(z) extends outward from the largest pole, or is the entire Z-plane except possibly $z=0$.

Note that $H(z)$ alone is not sufficient to determine causality; because it's not already sufficient to find the impulse response $h[n]$. Different impulse responses correspond to the same $H(z)$, for different ROCs.

Furthermore, since $H(z)$ of an LTI system uniquely specifies the corresponding LCCDE of the same system, therefore, in general for recursive systems, LCCDE alone is not sufficient to conclude for the causality of the corresponding system. You need additional information stated explicitly.

For example the LCCDE : $$ y[n] = 0.5 y[n-1] + x[n]$$

has two possible solutions for $n>0$ and for $n<0$ with proper initial conditions in both cases. Note that the former solution will be causal whereas the latter solution will be non-causal.

Finally, a block diagram can provide a pictorial definition for an LCCDE, and from a block diagram alone, for recursive systems, it's not possible to determine if it corresponds to a causal system or not, unless additional information is specified.

Note that if the system is non-recursive, (such as an FIR LTI) then you can directly determine causality based on the I/O relation, or $h[n]$, or $H(z)$, or the block diagram alone. For such non-recursive systems, you do not need additional information required for recursive systems.

Finally in your example, you are given a Direct-Form-II implementation of an LCDDE, and the additional information should be in the form of $n>0$ or $n<0$ to determine the causality fo the system that it corresponds to, however if it's not specified, then assuming $n>0$ is the more natural choice and your system will become causal then.