transfer function and 'causal' signal - evaluate transfer function or use z-transform of input?

Question

From my studying difference equations and transfer functions, I understand that when a complex exponential input $x[n]=z_1^n$ is applied to an LTI system with transfer function $H(z)$, determining the output involves evaluating the transfer function at $z=z_1$, i.e. $H(z_1)$ which is a complex quantity that simply scales the input signal to produce the output. The textbook I have uses a sinusoid input to show the effect (the transfer function causes a magnitude and phase change to the input).

The answers to this question I asked a while ago made the point of recognizing the difference between $a^n$ and $a^n \cdot u[n]$. All of that was in the context of two sided signals and the bilateral z-transform.

In something I found online, in the context of causal signals and the unilateral z-transform, there was a question posed: "if $x[n]=(1/2)^n \cdot u[n] \to y[n]=\delta [n-2],$ what is the output for $x[n]=\cos(\frac{\pi}{3}n)?$", where the author determined the output by evaluating $H(z)$ determined from the first relationship at $z=e^{jn \pi /3}$. Since this is a 'causal' context, isn't that cosine really $x[n]=\cos(n\frac{\pi}{3}) \cdot u[n]$ and hence requires taking the z-transform of the input?

When can you just evaluate $H(z)$ at the value of interest, and when do you have to take the (unilateral) z-transform of the input?

Answer 1

For a discrete-time LTI system with transfer function $H(z)$, the response to $x[n]=z_1^n$ equals $y[n]=H(z_1)z_1^n$ if $z_1$ is inside the region of convergence of $H(z)$. This relationship holds whether or not the system is causal or stable.

From this relationship it follows that if the system is real-valued and stable, i.e., if its impulse response is real-valued and if $H(e^{j\omega})=|H(e^{j\omega})|e^{j\phi(\omega)}$ exists, the response to a sinusoidal input $x[n]=\cos(n\omega_0)$ is given by $y[n]=|H(e^{j\omega_0})|\cos[n\omega_0+\phi(\omega_0)]$.

I think your confusion comes from assuming something you call a "causal context". It doesn't matter if a system is causal or not for above relationships to hold. You can apply a sinusoidal input signal to a causal system, and as long as the system is real-valued and stable, the output will be given by the relationship mentioned above. If the input is a sinusoid switched on at a finite $n=n_0$, the system's response can be computed by solving the convolution sum or, equivalently, by using the $\mathcal{Z}$-transform.

Answer 2

I'm not sure that the notion of a "'causal' context" is a meaningful distinction.

The author is taking a shortcut between the z domain and the discrete time Fourier transform domain.

In your case you've got a system with a transfer function $H(z)$, and you've got a choice of two signals: $x_i(n) = \cos (\theta_0 n)$ or $x_f(n) = u(n)\left(\cos( \theta_0 n)\right)$.

The big difference here is that $x_i$ is of infinite extent in both directions, where $x_f$ at least has a finite beginning.

You can go one of two ways here.

One way is to recognize that if $H(z)$ is the transfer function of the system in the context of the z transform, then $H(e^{j\theta})$ is the transfer function of the system in the context of the discrete-time Fourier transform*. Then the response of the system to a sinusoid of infinite extent at frequency $\theta$ and phase $\phi$ is just $\phi H(e^{j\theta})$. This is nice, but you have to crank through all that Fourier stuff to prove it.

Another way is to start with $x_f(n) = u(n)\left(\cos( \theta_0 n)\right)$, take it's $z$ transform, multiply that by $H(z)$ and do all the partial fraction math**. You'll find that if $H(z)$ has an impulse response that settles out, you end up with $y(n) = \left | H(z) \right | \cos \left (\theta_0 n + \arg H(z) \right) + y_t(n)$, where the modes of $y_t(n)$ are the modes of $H(z)$, which means that over time it settles out and you're left with $\left | H(z) \right | \cos \left (\theta_0 n + \arg \left( H(z) \right) \right)$.

In other words, after the transient has settled, the responses to $x_i$ and $x_f$ are the same. Moreover, the response to the "uses the z transform" case is the same as the "uses the Fourier transform" case. If you're very careful about minding your p's and your q's you can observe that H(z) is a LTI system, and the "TI" in "LTI" means "time invariant". This means that you can use the z-domain analysis technique for a signal $x^{'}_f(n) = u(n + k) \cos (\theta_0 n)$. Then you can take the limit as $k \to \infty$ -- and get back to the discrete-time Fourer transform case.

* Not the DFT -- the DFT is of finite extent, while the DTFT is of infinite extent.

** You should work this out for yourself if you can, starting with examples of $H(z)$ that you know to be stable, and then if you're feeling your oats, with any general $H(z)$ that's assumed to be stable.