x(t) is input in a system and the output is y(t). Could someone please explain m
ID: 2080803 • Letter: X
Question
x(t) is input in a system and the output is y(t). Could someone please explain mathematically why the following are either causal or noncausal and linear or non-linear (i'm able to do so by looking at a plot of the functions). My main issue is why they're causal/noncausal and linear/non-linear and not whether they are or not.
(a) (t) u(t) 2u (t 2) y (t) w(t) (1 -e t) w (t 2) (2 2e (-2 (b) T (t) u(t) u (t 2) y(t) u (t) e' sin(50Tt) u(t t-2 sin (50m (t -2)) -2)e (c) ar(t) u(t) u (t 1) y(t) u(t) (1 e t) w (t 1) (1 -e (t+1) z (t) u (t) u (t -1)) (d) u (t) (1 t) w (t -1) (1 y(t)Explanation / Answer
In general, a filter is causal if its output at present time (nn) never depends on the input at future times (n+mnm, with m>0m0). Let us restrict to LTI filters (and we assume discrete time) - so that the filter is fully specified by an impulse response function h(n)hn. In that case, the above property can be concisely stated as follows:
A LTI discrete-time filter is causal iff h(n)=0hn0 for n<0n0
This motivates the definition of a causal signal.
A discrete-time function (signal, sequence) g(n)gn is causal iff g(n)=0gn0 for n<0n0
Notice that this later definition does not involve filters (it's just motivated by them). And notice that the two can be combined in:
A LTI discrete-time filter is causal iff its response function h(n)hn is a causal function.
In your assertion regarding the second link ("it is causal because both xnxn and ynyn are 00 for n<0n0") you seem to be confusing both meanings. To determine that a filter is causal, one does not look for the causality of inputs or ouputs (xnxn and ynyn) but for the causality of the response function hnhn
Further, instead of h(n)hn we can work with its Z-transform H(z)Hz; we have y[n]=x[n]h[n]Y(z)=H(z)X(z)ynxnhnYzHzXz. But, remember the relation "signal" "Z transform" is not one-to-one unless the ROC (region of convergence) is also specified. A single H(z)Hz can have several corresponding h[n]hn("anti-transform"), for different ROCs. Alternatively, instead of giving a ROC, we might be given a causal (or anticausal) condition. In particular, if we are given a (rational) H(z)Hz and we are told that h(n)hn is causal, then the ROC must extend outwards from the biggest pole. For an explanation of this, see any Signal Processing textbook, or here.
In your example, Y(z)=zX(z)YzzXz, so H(z)=zHzz. To analyze this you can reason in two ways:
1) In terms of zeros and poles. H(z)Hz has a zero at z=0z0 and a pole at infinity. Hence, because there is a single ROC (all the plane), and it extends inwards from the pole, hence there can be only one valid h(n)hn which must be anti-causal.
A stable discrete-time LTI system is described by the following difference equation:
y[n]y[n1]+Cy[n2]=x[n]
ynyn1Cyn2xn
where C is a real number. Determine the range of C so that
(a) the system is causal;
(b) the system is anti-causal;
(c) the system is non-causal (i.e., it has a two-sided impulse response).
It is straight forward to calculate the transfer function:
Y(z)z1Y(z)+Cz2Y(z)H(Z)=Y(z)X(z)=X(z)=11z1+Cz2
Yzz1YzCz2Yz Xz HZYzXz 11z1Cz2
We are given that the system is stable so the ROC must include the unit circle. Therefore there can not be a pole with magnitude 11.
With C=0C0, H(z)=11z1Hz11z1, there is a pole at 11 so that is not possible.
With C=2C2, H(z)=11z12z2=131+z1+2312z1Hz11z12z2131z12312z1, there is a pole at 11 so that is not possible.
From there, where do I go?
Ultimately this may factor into the form:
k11+a1z1+k21+a2z1
k11a1z1k21a2z1
Is that a causal transfer function or not? The inverse Z transform can yield both a causal and anti-causal impulse response function.
A system is said to be linear when it satisfies superposition and homogenate principles. Consider two systems with inputs as x1(t), x2(t), and outputs as y1(t), y2(t) respectively. Then, according to the superposition and homogenate principles,
T [a1 x1(t) + a2 x2(t)] = a1 T[x1(t)] + a2 T[x2(t)]
, T [a1 x1(t) + a2 x2(t)] = a1 y1(t) + a2 y2(t)
From the above expression, is clear that response of overall system is equal to response of individual system.
(I insist: here you could deduce from H(z)Hz that the filter was anti-causal; but normally you can't; say, if H(z)=z/(z1)Hzzz1 you'd have two possible h[n]hn, one causal, one anticausal).
2) Explicitly, formally. By inspection, H(z)=nh(n)zn=zh(n)=(n+1)
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