Causal Signal Processing

Motivation:

The Fourier Transform is not suitable for real-time analysis. The reason lies in its non-causal nature: it requires the entire signal sequence from the first sample to the last to determine the frequency spectrum. This requirement poses a challenge in real-time applications because, in practical scenarios, we may not have access to future samples when processing a signal at any given time. This inherent delay makes the Fourier Transform inadequate for scenarios where immediate, real-time frequency analysis is necessary, such as in streaming audio or control systems.

Causal Systems:

In a causal system, the output at any time ( t ) depends only on values of the input at the current time and past times (i.e., no dependence on future values). Mathematically, we define a system as causal if its impulse response ( h(t) ) satisfies:

$$h(t)=0\text{for}t<0$$

This means that the system does not “anticipate” future inputs; it only reacts to current and past inputs. In physical terms, this is aligned with the concept that an effect cannot precede its cause.

Why Causality Makes Sense in Convolution:

To understand why causality is a logical requirement in certain contexts, consider the convolution integral, which defines the output ( y(t) ) of a linear time-invariant (LTI) system in response to an input ( x(t) ) as:

$$y(t)={\int }_{-\infty }^{\infty }h(t-\tau )x(\tau )d\tau$$ $$y(n)=h(n)\ast x(n)=\sum _{n=-\infty }^{\infty }h(n)x(m-n)$$

For ( y(t) ) to be causal, both the impulse response ( $h(t-\tau )$ ) and the input signal ( $x(\tau )$ ) must also be causal. If ( $h(t)\ne 0$ ) for ( $t<0$ ), the system would depend on future values of the input ( $x(t)$ ), violating causality. Thus, a causal ( $h(t)$ ) guarantees that the system response ( y(t) ) depends only on the present and past values of the input ( x(t) ), preserving causality.

Filtering and Convolution:

In signal processing, filtering is often defined as applying a convolution operation to an input signal to produce an output. Specifically, if we convolve an input signal ( x(t) ) with a filter’s impulse response ( h(t) ), we achieve the desired filtering effect in the output signal ( y(t) ):

$$y(t)=x(t)\ast h(t)$$

where ( * ) denotes the convolution operation. For real-time (causal) filtering applications, it is essential that both ( h(t) ) and ( x(t) ) are causal. Otherwise, the filtering operation would require future values of ( x(t) ), which are not available in real-time, thus making it impractical for live applications.

What is important to note is that: $Y(z)=X(z)\cdot H(z)$

Laplace Transform

The fourier transform isnt suitable for causal systems, because it requires the whole signal from $-\infty \to \infty$ , so we require a transform that works with continuous causal signals, which is where we use the Laplace transform

The problem however is that typically we aren’t using continous but sampled signals, which is why we use: Z-transform

How to Characterise Filters?

Transfer Function

Impulse Response