Electronics Guide

Fixed-Point Number Representation

A fixed-point number is an integer interpreted as a fraction by associating with it an implied scaling factor, almost always a power of two. The wordlength of the format is partitioned conceptually into a sign bit, a set of integer bits to the left of the implied radix point, and a set of fractional bits to its right. Because the radix point sits at a fixed position determined by the format rather than stored with each value, fixed-point arithmetic reduces to ordinary integer arithmetic with bookkeeping of the scaling, which is why it executes so efficiently on simple hardware.

Signed fixed-point values are represented almost universally in two's complement form, in which the most significant bit carries negative weight. Two's complement has the practical virtue that addition and subtraction proceed identically for signed and unsigned operands, requiring no special handling of the sign, and that a sequence of additions yielding a final result within range produces the correct answer even if intermediate partial sums overflow and wrap. This last property is exploited deliberately in some accumulation schemes, as later sections describe.

The uniform spacing of fixed-point values is the defining contrast with floating-point representation. Every representable fixed-point number is separated from its neighbors by the same quantization step, equal to the weight of the least significant bit. Floating-point numbers, by contrast, space themselves logarithmically, maintaining roughly constant relative precision across an enormous range. Fixed-point precision is therefore uniform in absolute terms but varies in relative terms, being fine for large-magnitude signals and coarse for small ones, which makes scaling decisions central to fixed-point design.

The range and resolution of a fixed-point format are fixed once the wordlength and radix-point position are chosen. Adding integer bits extends the range at which overflow occurs but, for a fixed total wordlength, removes fractional bits and coarsens the resolution. This direct trade between range and precision, absent in floating-point arithmetic, is the fundamental tension the fixed-point designer manages, and it recurs in every decision about formats, scaling, and wordlength allocation throughout a filter.

Q-Format Notation

Q-format notation is the standard shorthand for specifying where the implied radix point lies within a fixed-point word. In the common two-number form, a format written as Qm.n designates m integer bits and n fractional bits, so that the least significant bit carries a weight of two raised to the negative n. A frequently used single-number variant writes only the fractional count, as in Q15, leaving the integer and sign bits to be inferred from the wordlength. The notation makes explicit the scaling that the bare integer value alone does not convey.

The Q15 format exemplifies fractional fixed-point representation in sixteen-bit systems. One sign bit and fifteen fractional bits represent values in the range from negative one up to just below positive one, with a resolution of two raised to the negative fifteenth power. This pure-fractional convention is favored for signal samples and filter coefficients because the product of two fractions remains a fraction, so multiplication does not increase the integer range and the result stays naturally bounded near unity magnitude.

Arithmetic on Q-format numbers requires tracking how the format changes through each operation. Adding two numbers of the same Q-format yields a result in that same format, but the sum may need an additional integer bit to accommodate growth in magnitude. Multiplying a Qm.n value by a Qp.q value produces a result in Q(m+p).(n+q) format, doubling the fractional bits and widening the word, which is why a product of two sixteen-bit Q15 values occupies a thirty-two-bit Q30 result before any rounding or rescaling.

Disciplined use of Q-format notation prevents the scaling errors that otherwise plague fixed-point code. Each variable and each intermediate result carries a definite format, and the programmer or hardware designer aligns radix points before addition and accounts for the format change after multiplication. Tracking these formats explicitly, whether by hand, by comment convention, or through tools that propagate them automatically, turns the otherwise error-prone management of implied scaling into a systematic and verifiable process.

Coefficient Quantization

Filter design produces coefficients as high-precision real numbers, but implementation must round each to the nearest value representable in the chosen fixed-point format. This coefficient quantization perturbs the filter away from its designed transfer function, displacing the poles and zeros from their intended locations. Because the realized filter uses the quantized coefficients, its frequency response is the response of a slightly different filter, and the discrepancy grows as the coefficient wordlength shrinks.

The consequences differ between the zeros and the poles. Misplaced zeros, characteristic of both FIR and IIR filters, typically raise the stopband floor and introduce ripple, because the response no longer reaches the deep nulls the design specified. Misplaced poles, present only in recursive IIR filters, are far more dangerous: a pole driven outside the unit circle by coefficient quantization renders the filter unstable, and even a pole that merely drifts toward the unit circle can sharpen a resonance or narrow a band beyond the design intent.

Sensitivity to coefficient quantization depends strongly on the filter structure and on the proximity of poles to the unit circle. A high-order direct-form IIR filter is acutely sensitive, because every pole is determined jointly by all the denominator coefficients, so quantizing any one coefficient disturbs every pole at once. Decomposing the filter into a cascade of second-order sections confines each coefficient's influence to a single section's pair of poles, sharply reducing sensitivity and making the cascade form the standard remedy for coefficient quantization in high-order recursive filters.

The designer addresses coefficient quantization by allocating sufficient coefficient wordlength and by choosing a structure that tolerates the available precision. Coefficient wordlength can be set independently of signal wordlength, and a few additional coefficient bits often resolve a marginal pole placement at negligible cost. Where poles lie very close to the unit circle, as in narrowband and high-Q designs, specialized structures that represent the pole positions more robustly, or that store coefficients in transformed form, preserve the intended response under coarse quantization better than the straightforward direct form.

Signal Scaling and Overflow

Overflow occurs when a computed value exceeds the range the fixed-point format can represent, and in a filter the values most at risk are the internal sums rather than the input or output. The accumulation of products in a filter section can attain a magnitude well above that of any single sample, so a signal comfortably within range at the input may drive an internal node past the representable limit. Preventing this requires scaling the computation so that internal signals remain bounded throughout the filter.

Scaling is the deliberate introduction of gain factors, ordinarily powers of two implemented as bit shifts, that reduce signal levels at chosen points so internal nodes stay within range. The discipline trades headroom against precision: scaling down to guarantee no overflow lowers the signal relative to the fixed quantization noise floor, reducing the signal-to-noise ratio, whereas scaling too little courts overflow. The art of fixed-point scaling lies in choosing factors that hold overflow to an acceptable probability while preserving as much dynamic range as the application demands.

Several scaling criteria balance this trade-off differently. The most conservative, absolute-sum or L-one scaling, guarantees that overflow can never occur for any bounded input by bounding each node by the sum of the absolute values of the impulse response from input to that node, but it is pessimistic and sacrifices considerable signal-to-noise ratio. Less conservative criteria, such as scaling by the peak gain of the frequency response or by the energy of the impulse response, permit higher signal levels and better noise performance at the cost of a small, quantified probability of overflow for worst-case signals.

In multi-section filters the placement of scaling factors interacts with the ordering of the sections. A cascade of biquads requires scaling at the input of each section so that the signal entering it remains bounded, and the order in which sections appear affects how signal level builds through the chain. Practical designs order the sections and distribute the scaling jointly, placing high-gain resonant sections where the signal level has been reduced and balancing the dynamic range across the cascade to maximize the overall signal-to-noise ratio.

Rounding and Truncation Noise

Every fixed-point multiplication produces a result wider than its operands, and reducing this result back to the working wordlength discards low-order bits, introducing quantization error. The reduction is performed either by truncation, which simply drops the surplus bits, or by rounding, which adds a half-step before dropping them so the retained value is the nearest representable number. Both operations inject an error at each multiplication, and the accumulation of these errors establishes the noise floor of the filter output.

Rounding and truncation differ in the statistics of the error they produce. Rounding yields an error symmetric about zero, with no average bias, and is therefore preferred wherever the small additional hardware to add the rounding constant can be afforded. Truncation of two's complement numbers produces an error that is always of one sign, introducing a small negative bias, sometimes called a direct-current offset, that can accumulate through a recursive filter and that matters in applications sensitive to a constant output shift.

Quantization noise is commonly analyzed by modeling each rounding error as an independent random variable spread uniformly over one quantization step, an approximation that holds well when signals are busy and the step is small. Under this model the noise injected at each multiplier has a known variance proportional to the square of the least significant bit, and the contributions from all the multipliers can be summed to predict the total output noise. The model permits the signal-to-quantization-noise ratio of a proposed design to be estimated before implementation.

The structure of the filter governs how injected noise propagates to the output, and recursive filters differ fundamentally from non-recursive ones. In an FIR filter the rounding errors travel directly to the output and add independently, giving a noise floor that is simply their sum. In an IIR filter the errors injected within the feedback loop are themselves filtered by the recursive section, so noise at frequencies near a resonant pole is amplified, and the output noise depends jointly on the structure, the scaling, and the pole locations. This noise feedback makes IIR noise analysis more involved and the choice of structure more consequential.

Limit Cycles

Limit cycles are small, self-sustaining oscillations that appear in recursive fixed-point filters as a consequence of the nonlinearity that quantization introduces into the feedback loop. Because the quantizer rounds each recirculating value to the nearest representable level, the effective feedback is no longer exactly linear, and this nonlinearity can trap the filter in a periodic pattern that persists indefinitely. The phenomenon has no counterpart in FIR filters, which lack the feedback through which such a pattern could sustain itself.

Zero-input limit cycles are the most striking manifestation: a filter that should decay to silence once its input ceases instead settles into a steady oscillation or a fixed nonzero level. The effect arises because rounding can prevent the recirculating signal from decreasing below a certain magnitude, so the value that ought to decay toward zero is instead held at a small bound, where the feedback maintains it. The amplitude of such a limit cycle is on the order of a few least significant bits, defining an effective dead band within which the filter cannot settle to true zero.

A second class, overflow limit cycles, is far more severe and stems from the wraparound behavior of two's complement overflow rather than from rounding. When an internal sum overflows and wraps from a large positive value to a large negative one, the abrupt sign reversal inside a feedback loop can sustain a large-amplitude oscillation that swings across the full range of the representation. Unlike the small rounding limit cycles, overflow oscillations can dominate the output entirely and constitute a catastrophic failure of the filter.

Several techniques suppress or eliminate limit cycles. Saturation arithmetic, by clipping rather than wrapping overflowed values, removes the sign reversal that drives overflow oscillations and is the primary defense against them. Small rounding limit cycles are mitigated by adding controlled dither, by employing rounding schemes such as magnitude truncation that steer the recirculating value toward zero, or by adopting structures such as certain error-feedback and wave digital realizations that are provably free of zero-input limit cycles. The choice among these measures weighs the residual oscillation an application can tolerate against the cost of the countermeasure.

Dynamic Range and Wordlength

Dynamic range is the ratio between the largest and smallest signal magnitudes a system can represent with acceptable fidelity, and in a fixed-point system it is set directly by the wordlength. Each additional bit doubles the number of representable levels and so extends the dynamic range by approximately six decibels, the familiar rule that relates bits to the signal-to-quantization-noise ratio of a full-scale signal. Choosing the wordlength therefore amounts to choosing how much dynamic range the implementation provides.

The usable dynamic range of a filter is squeezed from both ends. At the top, the need to scale internal signals down to prevent overflow lowers the largest level the filter actually carries, sacrificing headroom that the format nominally offers. At the bottom, the quantization noise floor sets the smallest signal that remains distinguishable from rounding error. The dynamic range available to the signal is the span between these limits, narrower than the raw range of the wordlength and dependent on the scaling discipline the designer adopts.

Wordlength need not be uniform throughout a filter, and good designs assign precision where it yields the greatest benefit. Coefficient wordlength is chosen to place poles and zeros accurately enough to meet the response specification; signal wordlength is chosen to keep the noise floor acceptably low; and accumulator wordlength is made generous, with extra guard bits, because precision is cheapest to provide at the point of accumulation. Tailoring these wordlengths independently lets a design meet its requirements without paying for precision where it is not needed.

Selecting wordlengths is ultimately an exercise in matching the implementation to the application's true requirements. An audio filter may demand the dynamic range of sixteen or twenty-four bits to render quiet passages without audible noise, whereas a control loop may meet its goals with far fewer bits. Simulation of the fixed-point filter against representative signals, comparing its output to the infinite-precision reference, verifies that the chosen wordlengths deliver the required accuracy before the design is committed to hardware or fixed-point code.

Saturation Arithmetic

Saturation arithmetic changes the response to overflow from wraparound to clipping: a result that exceeds the representable range is replaced by the nearest range limit, the most positive value for positive overflow and the most negative value for negative overflow. This behavior mirrors the graceful clipping of an analog amplifier driven beyond its rails, in which excess signal is flattened rather than inverted, and it stands in sharp contrast to the abrupt sign reversal of unguarded two's complement overflow.

The decisive advantage of saturation appears within feedback loops. Two's complement wraparound transforms a large positive sum into a large negative one, and this sudden reversal, recirculated through a recursive filter, can ignite a destructive overflow limit cycle that swings across the entire range. Saturation eliminates the reversal, holding the overflowed value at the boundary, so the filter recovers as the signal subsides instead of locking into a large oscillation. Saturation is consequently regarded as essential for robust fixed-point IIR filters.

Saturation is not without cost, since clipping is itself a nonlinear operation that distorts the signal whenever it engages. A saturated sample no longer bears a linear relationship to the input, and frequent saturation injects harmonic distortion and intermodulation products into the output. The aim of scaling is therefore to make saturation a rare safety net rather than a routine occurrence: the signal levels are set so that the arithmetic seldom reaches the limits, and saturation merely contains the occasional worst-case excursion that scaling alone cannot guarantee to prevent.

Practical processors and hardware support saturation efficiently, often as a selectable mode of the arithmetic unit or as dedicated saturating instructions, so that clipping incurs little or no runtime penalty. Many digital signal processors also provide a wide accumulator with guard bits that absorbs the growth of a sum during a multiply-accumulate sequence and saturates only when the final result is stored back to the working wordlength. This arrangement combines the overflow protection of saturation with the noise advantage of accumulating at full precision and rounding once.

Summary

Fixed-point implementation trades the convenience of floating-point arithmetic for efficiency, accepting in return a set of quantization effects that the designer must understand and control. Numbers are represented as scaled integers in two's complement form, their radix points specified by Q-format notation that must be tracked through every addition and multiplication. Coefficient quantization displaces poles and zeros, degrading the response and threatening the stability of recursive filters, which the cascade decomposition into second-order sections largely tames. Signal scaling holds internal sums within range, balancing overflow against the loss of signal-to-noise ratio, while rounding and truncation set the noise floor that the filter structure shapes on its way to the output.

The recursive structure of IIR filters introduces effects absent from FIR designs, most notably limit cycles, the small zero-input oscillations of rounding and the catastrophic oscillations of overflow, which disciplined rounding and saturation arithmetic suppress. Dynamic range and wordlength tie these concerns together, the available bits allocated independently to coefficients, signals, and accumulators so that precision is spent where it counts. Mastering these interacting effects, through careful representation, scaling, arithmetic choices, and verification against an infinite-precision reference, is what allows a fixed-point filter to deliver, in compact and economical hardware, the response its design promised.

Related Topics

• Digital Filter Implementation
• IIR Filter Structures
• FIR Filter Structures
• Computer Arithmetic and Algorithms
• Digital Signal Processing Hardware