Impulse Invariance and Pole Mapping: Analog to Digital

1. Why Map from Analog to Digital?

Before we delve into the mathematics, it's worth asking why we would want to convert an analog filter design into a digital one. Why not just design digital filters from scratch? As you appreciate the historical context behind concepts, there are two primary motivations.

As the video explains, the two key drivers are:

1. Leveraging a Rich Heritage: The theory and design of analog filters (like Butterworth, Chebyshev, and Elliptic filters, which we will study later) were developed and refined over many decades. It is often more efficient to adapt these proven designs to the digital domain than to start from a blank slate.
2. Simulation: To simulate a continuous-time physical system on a computer, we must create a discrete-time model that accurately represents it. This requires a principled way to convert the system's continuous-time description to a discrete-time one.

2. The Principle of Impulse Invariance

The impulse invariance method is one of the most intuitive approaches to this conversion. The core idea is to create a digital filter whose impulse response is a sampled version of the original analog filter's impulse response.

Specifically, if the analog filter has an impulse response $h_a(t)$, we want the digital filter's impulse response, $h_d[n]$, to be "invariant" at the sampling instants.
$h_d[n]=h_a(nT)$
Here, $T$ is the sampling period. By preserving the shape of the impulse response at the sampling points, the hope is to also preserve the filter's characteristics.

3. Deriving the Pole Mapping

Now, let's perform the derivation, which will take us from this simple time-domain relationship to a direct mapping of poles in the complex plane. This process elegantly combines the Laplace and z-transforms.

The derivation follows these steps:

1. Start with a general analog transfer function, $H_a(s)$, and expand it into first-order terms using a partial fraction expansion.
2. Take the inverse Laplace transform to find the continuous-time impulse response, $h_a(t)$.
3. Sample $h_a(t)$ at intervals of $T$ to get the discrete-time impulse response, $h_d[n]$.
4. Take the z-transform of $h_d[n]$ to find the digital transfer function, $H_d(z)$.
5. Compare the forms of $H_a(s)$ and $H_d(z)$ to see how the poles and residues have been transformed.

The following resource provides a clear walkthrough of this mathematical process.

Let's summarize the key steps from that derivation.

We start with the analog transfer function $H_a(s)$ expressed in its partial fraction expansion, assuming it has $N$ distinct poles $s_k$ with corresponding residues $A_k$:
$H_a(s)={\sum }_{k=1}^N\frac{A_k}{s-s_k}$

The inverse Laplace transform gives us the causal analog impulse response:
$h_a(t)={\sum }_{k=1}^N{A}_k{e}^{s_{k}t}u(t)$

Next, we sample this response at $t=nT$, which is the defining step of impulse invariance:
$h_d[n]=h_a(nT)={\sum }_{k=1}^N{A}_k{e}^{s_{k}nT}u[n]$
We can rewrite this as:
$h_d[n]={\sum }_{k=1}^N{A}_k(e^{s_{k}T}{)}^{n}u[n]$

Finally, we take the z-transform. Recognizing the form of a geometric series ($\mathcal{Z}\{a^{n}u[n]\}=\frac{1}{1-a{z}^{-1}}$), we get the digital transfer function $H_d(z)$:
$H_d(z)={\sum }_{k=1}^N{A}_k\frac{1}{1-(e^{s_{k}T})z^{-1}}$

By comparing the initial and final forms, we arrive at the central conclusion:

• An analog pole at $s=s_k$ is mapped to a digital pole at $z=e^{s_{k}T}$.
• The residues $A_k$ are preserved in the partial fraction expansion.

This elegant result forms the heart of the impulse invariance method.

4. Implications of the Pole Mapping

The relationship $z_k=e^{s_{k}T}$ has profound consequences for both stability and the frequency response.

Stability Preservation

As we saw in the previous lesson, a causal analog system is stable if all its poles $s_k={\sigma }_k+j{\Omega }_k$ have a negative real part, ${\sigma }_k<0$. Let's see how this translates to the z-plane.

The magnitude of the corresponding digital pole $z_k$ is:
$|z_k|=|e^{s_{k}T}|=|e^{({\sigma }_k+j{\Omega }_k)T}|=|e^{{\sigma }_{k}T}{e}^{j{\Omega }_{k}T}|=|e^{{\sigma }_{k}T}|\cdot |e^{j{\Omega }_{k}T}|=e^{{\sigma }_{k}T}$
Since $|e^{j\theta }|=1$.

If the analog filter is stable, then ${\sigma }_k<0$. Because $T$ is always positive, ${\sigma }_{k}T<0$, which means $e^{{\sigma }_{k}T}<1$. Therefore, $|z_k|<1$.

This proves a vital property of impulse invariance: it always maps a stable analog filter to a stable digital filter. The stable left half of the s-plane is mapped to the interior of the unit circle in the z-plane, perfectly aligning with the stability criteria we established in the last lesson.

Frequency Response and Aliasing

While the pole mapping is direct and stability is preserved, the effect on the frequency response is more complex. Recall from the sampling theorem that when we sample a signal, its spectrum becomes periodic. The same happens here, but with the filter's frequency response. The relationship is:
$H_d(e^{j\omega })=\frac{1}{T}{\sum }_{k=-\infty }^{\infty }{H}_a\left(j\frac{\omega -2\pi k}{T}\right)$
where $\omega =\Omega T$ is the normalized discrete-time frequency.

This equation tells us that the digital filter's frequency response is a superposition of scaled and shifted replicas of the analog filter's frequency response. If the analog filter is not perfectly band-limited (i.e., $H_a(j\Omega )\ne 0$ for $|\Omega |>\pi /T$), these replicas will overlap, causing aliasing.

This aliasing is the primary drawback of the impulse invariance method. It distorts the frequency response, especially near the Nyquist frequency ($\omega =\pi$). The method works best for analog filters that are already mostly band-limited, such as sharp low-pass or band-pass filters, where the response is negligible outside the main band, minimizing the overlap.

Visualizing the Mapping

To build a deeper intuition for how continuous-time system characteristics translate to the discrete-time domain, it is useful to see how entire regions and lines in the s-plane are mapped. Given your background in control systems, visualizing how things like damping and natural frequency are transformed is particularly insightful.

Conclusion

In this lesson, we have formally bridged the gap between analog and digital systems by deriving the principles of the impulse invariance method. We saw how a simple and intuitive time-domain requirement—preserving the impulse response at sampling instants—leads to a direct and powerful relationship in the complex frequency domains.

Key Takeaways:

• Impulse Invariance: A technique to design a digital filter by sampling the impulse response of an analog prototype, i.e., $h_d[n]=h_a(nT)$.
• Pole Mapping: The core result of this method is that an analog pole at $s_k$ is mapped to a digital pole at $z_k=e^{s_{k}T}$. The residues of the partial fraction expansion are preserved.
• Stability Preservation: This pole mapping ensures that a stable analog filter (poles in the left-half s-plane) will always result in a stable digital filter (poles inside the unit circle).
• Limitation: Aliasing: The main drawback is that the frequency response of the digital filter suffers from aliasing unless the analog filter is perfectly band-limited, which is never the case in practice. This limits its use to filters where out-of-band energy is minimal.

Preview of the Next Lesson:

We have now concluded our foundational exploration of discrete-time signals and systems in the z-domain. We've covered the z-transform, its properties, stability analysis, and now a method for filter design. The next major topic in our course is the analysis of signals in the frequency domain. While the DTFT provides the theoretical framework, it is a continuous function of frequency and cannot be computed directly.

In our next module, we will introduce its practical, computable counterpart: the Discrete Fourier Transform (DFT). We will begin by tracing the fascinating history of the algorithm that makes it efficient—the Fast Fourier Transform (FFT)—from its surprising origins with Gauss in the early 19th century to its modern rediscovery by Cooley and Tukey.