In case of recursive system, a feedback connection is present. So if there is an overflow then it is feedback and used to come to the next output, where it causes further overflow.

This creates undesired oscillations at the output, hence results in nonlinearity. It becomes difficult to analyze the digital filter precisely.

To limit this overflow it is required to scale the input signal and unit sample response. This scaling is done between the input and any internal summing node in the system.

Let us assume that YK denotes the response of the system at kth node, for the input x(n) and let hk(n) be the impulse response of the system.

According to the definition of convolution:

$y_k (n)=∑_{k=-∞}^∞h_k (m) . x(n-m)$

Taking magnitudes of both sides;

$|y_k (n)|=|∑_{k=-∞}^∞h_k (m) . x(n-m)|$

$|y_k (n)|≤A_x ∑_{m=-∞}^∞|h_k (m)|$

$A_x≤\frac{1}{(∑_{m=-∞}^∞|h_k (m)|)}$

This is the necessary and sufficient condition to prevent overflow in the system. It means to avoid the overflow, proper dynamic scaling should be done at that particular node.