As far as I know, memoryless systems are causal systems. But why aren't systems with memory necessarily causal?
Since the system with memory is affected by past input and current input, I think that this property is the same as causality.
Signal Processing Stack Exchange is a question and answer site for practitioners of the art and science of signal, image and video processing. It only takes a minute to sign up.
Sign up to join this communityAs far as I know, memoryless systems are causal systems. But why aren't systems with memory necessarily causal?
Since the system with memory is affected by past input and current input, I think that this property is the same as causality.
A system is memoryless if its output at a given time is dependent only on the input at that same time (and potentially the time itself). The converse is called a system with memory ("memory system" or "non-memoryless"): it can use past or future information. A causal system only on past inputs and outputs.
Nota: the notion of "future" here is relative to the system. To me, it amount to getting/buffering future samples prior to computing something on the present sample. One example is the decoding on images in video. The compression principle being based on predicting motions from one frame to the others, standards use intra pictures (coded independently), and others in a "GoP" (group of picture) can be coded using past or past and future images, resulting in a non causal transmissions, but with memory.
There other pointers:
A system with memory might depend its output at older input.
It doesn't say it can't also depend on future input.
So, "having memory" and "being causal" are simply two different things.
Having memory and being causal are two different things. However, there is some correlation between the two, because a memoryless (also called instantaneous) system must be causal because it only depends on the current input, not on any past or future inputs.
A system with memory (also called dynamic) can be either causal or non-causal. This is simply the case because any system that is not instantaneous must be dynamic, i.e., have memory, which, by definition, must include causal and non-causal systems.
E.g., the system with input-output relation
$$y(t)=\int_{t-T}^{t+T}x(\tau)d\tau\tag{1}$$
is certainly not memoryless, hence it is dynamic. It depends on past inputs as well as on future inputs, making it a non-causal system.