Impulse response and convolution The behavior of a linear, continuous-time, time-invariant system with input signal
x(
t) and output signal
y(
t) is described by the convolution integral: : where h(t) is the system's response to an
impulse: x(\tau) = \delta(\tau). y(t) is therefore proportional to a weighted average of the input function x(\tau). The weighting function is h(-\tau), simply shifted by amount t. As t changes, the weighting function emphasizes different parts of the input function. When h(\tau) is zero for all negative \tau, y(t) depends only on values of x prior to time t, and the system is said to be
causal. To understand why the convolution produces the output of an LTI system, let the notation \{x(u-\tau);\ u\} represent the function x(u-\tau) with variable u and constant \tau. And let the shorter notation \{x\} represent \{x(u);\ u\}. Then a continuous-time system transforms an input function, \{x\}, into an output function, \{y\}. And in general, every value of the output can depend on every value of the input. This concept is represented by: y(t) \mathrel{\stackrel{\text{def}}{=}} O_t\{x\}, where O_t is the transformation operator for time t. In a typical system, y(t) depends most heavily on the values of x that occurred near time t. Unless the transform itself changes with t, the output function is just constant, and the system is uninteresting. For a linear system, O must satisfy : {{NumBlk|:| O_t\left\{\int\limits_{-\infty}^\infty c_{\tau}\ x_{\tau}(u) \, \mathrm{d}\tau ;\ u\right\} = \int\limits_{-\infty}^\infty c_\tau\ \underbrace{y_\tau(t)}_{O_t\{x_\tau\}} \, \mathrm{d}\tau. |}} And the time-invariance requirement is: {{NumBlk|:| \begin{align} O_t\{x(u - \tau);\ u\} &\mathrel{\stackrel{\quad}{=}} y(t - \tau)\\ &\mathrel{\stackrel{\text{def}}{=}} O_{t-\tau}\{x\}.\, \end{align} }} In this notation, we can write the
impulse response as h(t) \mathrel{\stackrel{\text{def}}{=}} O_t\{\delta(u);\ u\}. Similarly: : Substituting this result into the convolution integral: \begin{align} (x * h)(t) &= \int_{-\infty}^\infty x(\tau)\cdot h(t - \tau) \,\mathrm{d}\tau \\[4pt] &= \int_{-\infty}^\infty x(\tau)\cdot O_t\{\delta(u-\tau);\ u\} \, \mathrm{d}\tau,\, \end{align} which has the form of the right side of for the case c_\tau = x(\tau) and x_\tau(u) = \delta(u-\tau). then allows this continuation: \begin{align} (x * h)(t) &= O_t\left\{\int_{-\infty}^\infty x(\tau)\cdot \delta(u-\tau) \, \mathrm{d}\tau;\ u \right\}\\[4pt] &= O_t\left\{x(u);\ u \right\}\\ &\mathrel{\stackrel{\text{def}}{=}} y(t).\, \end{align} In summary, the input function, \{x\}, can be represented by a continuum of time-shifted impulse functions, combined "linearly", as shown at . The system's linearity property allows the system's response to be represented by the corresponding continuum of impulse responses, combined in the same way. And the time-invariance property allows that combination to be represented by the convolution integral. The mathematical operations above have a simple graphical simulation.
Exponentials as eigenfunctions An
eigenfunction is a function for which the output of the operator is a scaled version of the same function. That is, \mathcal{H}f = \lambda f, where
f is the eigenfunction and \lambda is the
eigenvalue, a constant. The
exponential functions A e^{s t}, where A, s \in \mathbb{C}, are
eigenfunctions of a
linear,
time-invariant operator. A simple proof illustrates this concept. Suppose the input is x(t) = A e^{s t}. The output of the system with impulse response h(t) is then \int_{-\infty}^\infty h(t - \tau) A e^{s \tau}\, \mathrm{d} \tau which, by the
commutative property of
convolution, is equivalent to \begin{align} \overbrace{\int_{-\infty}^\infty h(\tau) \, A e^{s (t - \tau)} \, \mathrm{d} \tau}^{\mathcal{H} f} &= \int_{-\infty}^\infty h(\tau) \, A e^{s t} e^{-s \tau} \, \mathrm{d} \tau \\[4pt] &= A e^{s t} \int_{-\infty}^{\infty} h(\tau) \, e^{-s \tau} \, \mathrm{d} \tau \\[4pt] &= \overbrace{\underbrace{A e^{s t}}_{\text{Input}}}^{f} \, \overbrace{\underbrace{H(s)}_{\text{Scalar}}}^{\lambda} \, , \\ \end{align} where the scalar H(s) \mathrel{\stackrel{\text{def}}{=}} \int_{-\infty}^\infty h(t) e^{-s t} \, \mathrm{d} t is dependent only on the parameter
s. So the system's response is a scaled version of the input. In particular, for any A, s \in \mathbb{C}, the system output is the product of the input A e^{st} and the constant H(s). Hence, A e^{s t} is an
eigenfunction of an LTI system, and the corresponding
eigenvalue is H(s).
Direct proof It is also possible to directly derive complex exponentials as eigenfunctions of LTI systems. Let's set v(t) = e^{i \omega t} some complex exponential and v_a(t) = e^{i \omega (t+a)} a time-shifted version of it. H[v_a](t) = e^{i\omega a} H[v](t) by linearity with respect to the constant e^{i \omega a}. H[v_a](t) = H[v](t+a) by time invariance of H. So H[v](t+a) = e^{i \omega a} H[v](t). Setting t = 0 and renaming we get: H[v](\tau) = e^{i\omega \tau} H[v](0) i.e. that a complex exponential e^{i \omega \tau} as input will give a complex exponential of same frequency as output.
Fourier and Laplace transforms The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems. The one-sided
Laplace transform H(s) \mathrel{\stackrel{\text{def}}{=}} \mathcal{L}\{h(t)\} \mathrel{\stackrel{\text{def}}{=}} \int_0^\infty h(t) e^{-s t} \, \mathrm{d} t is exactly the way to get the eigenvalues from the impulse response. Of particular interest are pure sinusoids (i.e., exponential functions of the form e^{j \omega t} where \omega \in \mathbb{R} and j \mathrel{\stackrel{\text{def}}{=}} \sqrt{-1}). The
Fourier transform H(j \omega) = \mathcal{F}\{h(t)\} gives the eigenvalues for pure complex sinusoids. Both of H(s) and H(j\omega) are called the
system function,
system response, or
transfer function. The Laplace transform is usually used in the context of one-sided signals, i.e. signals that are zero for all values of
t less than some value. Usually, this "start time" is set to zero, for convenience and
without loss of generality, with the transform integral being taken from zero to infinity (the transform shown above with lower limit of integration of negative infinity is formally known as the
bilateral Laplace transform). The Fourier transform is used for analyzing systems that process signals that are infinite in extent, such as modulated sinusoids, even though it cannot be directly applied to input and output signals that are not
square integrable. The Laplace transform actually works directly for these signals if they are zero before a start time, even if they are not square integrable, for stable systems. The Fourier transform is often applied to spectra of infinite signals via the
Wiener–Khinchin theorem even when Fourier transforms of the signals do not exist. Due to the convolution property of both of these transforms, the convolution that gives the output of the system can be transformed to a multiplication in the transform domain, given signals for which the transforms exist y(t) = (h*x)(t) \mathrel{\stackrel{\text{def}}{=}} \int_{-\infty}^\infty h(t - \tau) x(\tau) \, \mathrm{d} \tau \mathrel{\stackrel{\text{def}}{=}} \mathcal{L}^{-1}\{H(s)X(s)\}. One can use the system response directly to determine how any particular frequency component is handled by a system with that Laplace transform. If we evaluate the system response (Laplace transform of the impulse response) at complex frequency , where , we obtain |
H(
s)| which is the system gain for frequency
f. The relative phase shift between the output and input for that frequency component is likewise given by arg(
H(
s)).
Examples {{bulleted list • \frac{\mathrm{d}}{\mathrm{d}t} \left( c_1 x_1(t) + c_2 x_2(t) \right) = c_1 x'_1(t) + c_2 x'_2(t) (i.e., it is linear) • \frac{\mathrm{d}}{\mathrm{d}t} x(t-\tau) = x'(t-\tau) (i.e., it is time invariant) When the Laplace transform of the derivative is taken, it transforms to a simple multiplication by the Laplace variable
s. \mathcal{L}\left\{\frac{\mathrm{d}}{\mathrm{d}t}x(t)\right\} = s X(s) That the derivative has such a simple Laplace transform partly explains the utility of the transform. \mathcal{A}\left\{x(t)\right\} \mathrel{\stackrel{\text{def}}{=}} \int_{t-a}^{t+a} x(\lambda) \, \mathrm{d} \lambda. By the linearity of integration, \begin{align} \mathcal{A} \{c_1 x_1(t) + c_2 x_2(t)\} &= \int_{t-a}^{t+a} ( c_1 x_1(\lambda) + c_2 x_2(\lambda)) \, \mathrm{d} \lambda\\ &= c_1 \int_{t-a}^{t+a} x_1(\lambda) \, \mathrm{d} \lambda + c_2 \int_{t-a}^{t+a} x_2(\lambda) \, \mathrm{d} \lambda\\ &= c_1 \mathcal{A}\{x_1(t)\} + c_2 \mathcal{A} \{x_2(t) \}, \end{align} it is linear. Additionally, because \begin{align} \mathcal{A}\left\{x(t-\tau)\right\} &= \int_{t-a}^{t+a} x(\lambda-\tau) \, \mathrm{d} \lambda\\ &= \int_{(t-\tau)-a}^{(t-\tau)+a} x(\xi) \, \mathrm{d} \xi\\ &= \mathcal{A}\{x\}(t-\tau), \end{align} it is time invariant. In fact, \mathcal{A} can be written as a convolution with the
boxcar function \Pi(t). That is, \mathcal{A}\left\{x(t)\right\} = \int_{-\infty}^\infty \Pi\left(\frac{\lambda-t}{2a}\right) x(\lambda) \, \mathrm{d} \lambda, where the boxcar function \Pi(t) \mathrel{\stackrel{\text{def}}{=}} \begin{cases} 1 &\text{if } |t| \frac{1}{2}. \end{cases} }}
Important system properties Some of the most important properties of a system are causality and stability. Causality is a necessity for a
physical system whose independent variable is time, however this restriction is not present in other cases such as image processing.
Causality A system is causal if the output depends only on present and past, but not future inputs. A necessary and sufficient condition for causality is h(t) = 0 \quad \forall t where h(t) is the impulse response. It is not possible in general to determine causality from the
two-sided Laplace transform. However, when working in the time domain, one normally uses the
one-sided Laplace transform which requires causality.
Stability A system is
bounded-input, bounded-output stable (BIBO stable) if, for every bounded input, the output is finite. Mathematically, if every input satisfying \ \|x(t)\|_{\infty} leads to an output satisfying \ \|y(t)\|_{\infty} (that is, a finite
maximum absolute value of x(t) implies a finite maximum absolute value of y(t)), then the system is stable. A necessary and sufficient condition is that h(t), the impulse response, is in
L1 (has a finite L1 norm): \|h(t)\|_1 = \int_{-\infty}^\infty |h(t)| \, \mathrm{d}t In the frequency domain, the
region of convergence must contain the imaginary axis s = j\omega. As an example, the ideal
low-pass filter with impulse response equal to a
sinc function is not BIBO stable, because the sinc function does not have a finite L1 norm. Thus, for some bounded input, the output of the ideal low-pass filter is unbounded. In particular, if the input is zero for t and equal to a sinusoid at the
cut-off frequency for t > 0, then the output will be unbounded for all times other than the zero crossings. == Discrete-time systems ==