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A recursive system is a system in which current output depends on previous output(s) and input(s) but in non-recursive system current output does not depend on previous output(s).
A recursive system is a system in which current output depends on previous output(s) and input(s) but in non-recursive system current output does not depend on previous output(s).
The system with memory is not necessarily a recursive system. For example in FIR systems for input x[n] and output y[n] if we have
y[n] = 0.5(x[n]+x[n-1]) then the current output does not depend on previous output but depends on current input and previous input. But for IIR case like an accumulator
y[n] = y[n-1]+x[n], current output is depended on previous output as well as on current input ( generally current and previous inputs). So accumulator is a recursive system.
First example is a system with memory as we have x[n-1] with x[n] i-e we have stored x[n-1] and used that to compute y[n] but it is non-recursive as y[n] does not depend on y[n-1] or y[n-2] etc. Second one is recursive because y[n] depends on y[n-1].
First example is a system with memory as we have x[n-1] with x[n] i-e we have stored x[n-1] and used that to compute y[n] but it is non-recursive as y[n] does not depend on y[n-1] or y[n-2] etc. Second one is recursive because y[n] depends on y[n-1].
FIR filters can have recursive implementations. the most obvious is the simple moving sum (or moving average):
y[n] = x[n] + x[n-1] + x[n-2] + ... + x[n-511]
y[n] = y[n-1] + x[n] - x[n-512]
the first method is the direct method, the second works because y[n] and y[n-1] share many common terms.
FIR = Finite Impulse Response
and recursive systems have generally infinite impulse response. So, they cannot be implemented as FIR. Only systems having Finite Impulse response can be implemented using FIR filters.
FIR filters can have utilize recursion within the implementation. I show the most common example of this in my previous post. For the specific case of the moving average (or moving accumulation), each input is multiplied by 1, then summed. thus for a 4 tap filter h = [1 1 1 1]:
y[10] = x[10] + x[9] + x[8] + x[7]
y[11] = x[11] + x[10] + x[9] + x[8]
are two samples from the sequence. notice that:
y[11] = y[10] + x[11] - x[7]
or "the next value is the current value, plus the new input, minus the oldest input"
IIR filters must be recursive, and furthermore, the impulse response is infinitely long. while IIR filters are often strongly associated with recursive systems, it is actually the impulse response that gives them the name. As shown above, there are recursive systems that have finite impulse responses.
All IIR filters are recursive
not all recursive systems are IIR
So if you know a system is IIR, you know it has recursion. If you only know that a system is recursive, you must do further analysis to determine FIR/IIR.
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