Directform

Direct form refers to a class of digital filter realizations used to implement discrete-time linear time-invariant systems. In such realizations, the filter’s difference equation is mapped onto a network of delay elements, multipliers, and adders. Direct-form structures are among the most common and straightforward ways to implement finite impulse response (FIR) and infinite impulse response (IIR) filters in software and hardware.

Direct Form I realizes a digital filter by separating the numerator and denominator polynomials of the transfer function. For a causal IIR of order N, the difference equation is typically written as y[n] = -a1 y[n-1] - ... - aN y[n-N] + b0 x[n] + b1 x[n-1] + ... + bN x[n-N]. In Direct Form I, the corresponding hardware or software path uses two parallel delay lines: one for the input samples to the b-terms (feedforward path) and one for the past outputs to the a-terms (feedback path). This straightforward arrangement is easy to understand and implement.

Direct Form II reduces memory by combining the delays into a single chain of state elements. It realizes the same transfer function with a single set of N delay elements storing intermediate states, which are used by both the feedforward and feedback computations. This makes Direct Form II more memory-efficient and often faster in fixed-point hardware, but it can be more sensitive to coefficient quantization and numerical errors, especially for filters with large poles.

Transposed forms rearrange the signal flow so that the roles of the delay lines and multipliers are exchanged. Transposed Direct Form I and Transposed Direct Form II are commonly used in hardware implementations to improve numerical stability and to suit specific data-path architectures.

Applications of direct-form realizations span audio processing, communications, and real-time digital signal processing, where efficient, stable, and scalable filter implementations are essential.