Characterizing Analog LTI Systems

1. The Time-Domain View: Impulse Response and Convolution

The most fundamental way to characterize an LTI system is to see how it responds to the most elementary input imaginable: a perfect, instantaneous kick. In continuous-time, this is the Dirac delta function, $\delta (t)$.

The impulse response, denoted $h(t)$, is defined as the output of an LTI system when the input is a delta function:
$h(t)=\mathcal{T}\{\delta (t)\}$
where $\mathcal{T}$ represents the system transformation.

Why is this one specific response so important? Because, thanks to the properties of linearity and time-invariance, knowing $h(t)$ allows us to determine the system's output for any input signal.

The key insight, which you'll recognize from your work with random processes and signal processing, is that any arbitrary input signal $x(t)$ can be represented as an infinite sum (an integral) of scaled and time-shifted delta functions. This is formalized by the sifting property of the delta function:
$x(t)={\int }_{-\infty }^{\infty }x(\tau )\delta (t-\tau )d\tau$
This equation says that the value of $x$ at time $t$ is constructed by adding up weighted impulses from all moments in time $\tau$.

Now, let's see what happens when we feed this signal into an LTI system:
$y(t)=\mathcal{T}\{x(t)\}=\mathcal{T}\left\{{\int }_{-\infty }^{\infty }x(\tau )\delta (t-\tau )d\tau \right\}$

Because the system is linear, we can move the transformation operator inside the integral (which is a form of superposition):
$y(t)={\int }_{-\infty }^{\infty }x(\tau )\mathcal{T}\{\delta (t-\tau )\}d\tau$

Because the system is time-invariant, a shifted input $\delta (t-\tau )$ simply produces a shifted output $h(t-\tau )$. So, $\mathcal{T}\{\delta (t-\tau )\}=h(t-\tau )$. Substituting this back in gives us the final, powerful result:
$y(t)={\int }_{-\infty }^{\infty }x(\tau )h(t-\tau )d\tau$

This operation is the convolution integral, denoted as $y(t)=(x\ast h)(t)$. It tells us that the output of any LTI system is simply the convolution of the input signal with the system's impulse response. Therefore, $h(t)$ is a complete characterization of the system in the time domain.

To build a strong intuition for this process, the following video gives an excellent visual explanation of how an input signal can be viewed as a series of impulses and how the output is formed by summing the system's responses to each of these impulses.

For a more rigorous mathematical treatment that formalizes this derivation for both discrete and continuous time, the following reading is highly recommended.

2. The s-Domain View: The Transfer Function

While convolution is fundamental, it's a computationally intensive operation. This is where the transform methods we've been studying show their true power.

In the previous lesson, when solving a differential equation like $a{y}^{\prime \prime }(t)+b{y}^{\prime }(t)+cy(t)=x(t)$ with zero initial conditions, you would have arrived at an algebraic equation in the s-domain:
$(a{s}^2+bs+c)Y(s)=X(s)$
$\frac{Y(s)}{X(s)}=\frac{1}{a{s}^2+bs+c}$
This ratio, which relates the output transform to the input transform, is the system's transfer function, $H(s)$.

Formally, the transfer function $H(s)$ of an LTI system is defined as the ratio of the Laplace transform of the output, $Y(s)$, to the Laplace transform of the input, $X(s)$, assuming all initial conditions are zero (i.e., the system is in a "zero state").
$H(s)\equiv \frac{Y(s)}{X(s)}{|}_{\text{zero initial conditions}}$

The transfer function characterizes the system's intrinsic properties, independent of any specific input or initial state. Now, let's connect this back to the impulse response.

We know from the convolution property of the Laplace transform that:
$\mathcal{L}\{x(t)\ast h(t)\}=X(s)H(s)$
Since $y(t)=x(t)\ast h(t)$, we have $Y(s)=X(s)H(s)$, where $H(s)=\mathcal{L}\{h(t)\}$.

This reveals two crucial relationships:

1. The transfer function $H(s)$ is the Laplace transform of the impulse response $h(t)$.
2. Convolution in the time domain becomes simple multiplication in the s-domain.

This is a profound result. It means we can analyze LTI systems by transforming $x(t)$ and $h(t)$ to $X(s)$ and $H(s)$, performing a simple multiplication to get $Y(s)$, and then transforming back to find $y(t)$. This algebraic approach is often far simpler than performing a convolution integral.

The following video provides an excellent overview of what transfer functions are, how they are derived from physical systems, and why they are so useful in practice.

3. The Frequency Domain View: The Frequency Response

The third crucial characterization of an LTI system describes its behavior when the input is a pure, steady-state sinusoid. Given your background in signal processing and control systems, you know that understanding how a system responds to different frequencies is essential.

Let's consider an input that is a complex exponential: $x(t)=e^{j{\omega }_{0}t}$. These functions are the eigenfunctions of LTI systems. This means that when the input is an eigenfunction, the output is the same function, merely scaled by a complex constant (the eigenvalue). Let's prove this.

Using the convolution integral for the output $y(t)$:
$y(t)={\int }_{-\infty }^{\infty }h(\tau )x(t-\tau )d\tau ={\int }_{-\infty }^{\infty }h(\tau )e^{j{\omega }_0(t-\tau )}d\tau$
We can factor out the term that doesn't depend on the integration variable $\tau$:
$y(t)=e^{j{\omega }_{0}t}{\int }_{-\infty }^{\infty }h(\tau )e^{-j{\omega }_0\tau }d\tau$

Look closely at the integral. It is the definition of the Fourier Transform of $h(t)$. It is also precisely our definition of the bilateral Laplace transform of $h(t)$ evaluated at $s=j{\omega }_0$.
${\int }_{-\infty }^{\infty }h(\tau )e^{-j{\omega }_0\tau }d\tau =H(s){|}_{s=j{\omega }_0}=H(j{\omega }_0)$

Therefore, the output is:
$y(t)=H(j{\omega }_0)e^{j{\omega }_{0}t}$

The complex quantity $H(j\omega )$ is called the frequency response of the system. It tells us how the system modifies the input sinusoid at each frequency $\omega$.

• The magnitude $|H(j\omega )|$ is the gain of the system. It tells you how much the amplitude of the sinusoid is amplified or attenuated.
• The phase $\angle H(j\omega )$ is the phase shift of the system. It tells you how much the sinusoid is shifted in time.

The frequency response is simply the transfer function evaluated along the imaginary axis of the s-plane. This provides the critical link between the s-domain analysis (poles and zeros) and the practical frequency-domain behavior of a system (e.g., as a filter).

Conclusion

Today, we've connected three critical perspectives on LTI systems, forming a unified framework for analysis. This trinity of system descriptions is a cornerstone of signals and systems theory.

Key Takeaways:

• Impulse Response ($h(t)$): The system's time-domain signature. The output is found by convolving the input with $h(t)$: $y(t)=x(t)\ast h(t)$.
• Transfer Function ($H(s)$): The system's s-domain description. It is the Laplace transform of the impulse response, $H(s)=\mathcal{L}\{h(t)\}$. It turns convolution into multiplication: $Y(s)=X(s)H(s)$.
• Frequency Response ($H(j\omega )$): Describes the system's steady-state response to sinusoids. It is the transfer function evaluated on the imaginary axis, $H(j\omega )=H(s){|}_{s=j\omega }$. For an input $e^{j\omega t}$, the output is $H(j\omega )e^{j\omega t}$.

This diagram summarizes the core relationships between the time domain (left) and frequency/s-domain (right) for LTI systems. The impulse response h(t) and transfer function H(s) are a Laplace transform pair. Convolution in time corresponds to multiplication in the s-domain.

Preview of the Next Lesson:
We have just established that the frequency response $H(j\omega )$ is found by evaluating the transfer function $H(s)$ on the $j\omega$-axis. In our next lesson, we will explore this relationship in much greater detail. We will analyze how the locations of a system's poles and zeros in the s-plane directly shape the magnitude and phase of its frequency response, giving us a powerful graphical tool for system analysis and design.