• 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.