Fractional Delay Filter

Unlike integer delay lines that shift signals by whole sample periods, a fractional delay filter provides a continuously variable delay τ = k + μ, where k is the integer part and μ ∈ [0,1) is the fractional part. This sub-sample precision is essential for compensating for analog path mismatches, temperature-dependent propagation delays, and phase offsets in the feedback loop that are not integer multiples of the ADC clock period.

Lagrange interpolation is the most common design method for the Farrow sub-filters. For a delay μ, the ideal impulse response is a shifted sinc function: h[n] = sinc(n − μ). Lagrange interpolation approximates this by fitting an N-th order polynomial through N+1 samples. Key properties:

• Maximally Flat: Magnitude response is flat at DC, providing excellent low-frequency accuracy.
• Odd Order Preference: Odd N (e.g., N=3, 5, 7) yields better passband performance.
• Coefficient Formulas: Closed-form expressions exist for direct computation of sub-filter taps.

The usable bandwidth of a fractional delay filter is limited by its approximation error. Performance metrics include:

• Passband Ripple: Deviation from unity gain across the signal bandwidth. For a 100 MHz NR signal, ripple must typically stay below 0.01 dB.
• Phase Linearity: Deviation from the ideal linear phase slope corresponding to the desired delay μ.
• Design Trade-off: Higher filter order N extends the flat bandwidth but increases latency and computational cost. A 5th-order Lagrange filter typically achieves >80% of the Nyquist bandwidth with acceptable error.

In a digital predistortion system, the fractional delay filter sits in the reference path (or feedback path) to align the transmit waveform with the observed PA output before coefficient estimation. Misalignment by even 0.1 samples can severely degrade model accuracy:

• AM-AM/AM-PM Distortion: Time misalignment smears the nonlinear characteristic, causing the DPD model to learn a distorted inverse.
• EVM Floor: Residual timing error directly sets a floor on achievable EVM.
• Joint Estimation: Often combined with loop delay estimation algorithms that first find the integer delay via cross-correlation, leaving the fractional residual to the Farrow filter.

In adaptive DPD systems operating over varying temperatures and frequencies, the optimal fractional delay may drift. The Farrow structure enables smooth tracking:

• μ Update: A delay-locked loop or gradient-based algorithm adjusts μ to minimize the mean squared error between reference and aligned feedback.
• No Transient Switching: Unlike switching between discrete delay taps, varying μ continuously produces no glitches or spectral splatter.
• Multi-Band Extension: Separate Farrow filters can independently align concurrent bands in multi-band DPD architectures.

A polynomial interpolation method that constructs a unique polynomial passing through a set of known sample points, widely used as the basis for fractional delay filter design.

• Maximally flat magnitude response at DC
• Coefficients derived directly from the Lagrange basis polynomials
• Odd-order designs (e.g., 3rd, 5th) preferred for linear-phase symmetry
• Excellent passband performance for narrowband signals; degrades near Nyquist

The process of determining the integer and fractional sample delay between the transmitted reference signal and the observed feedback signal in a DPD system.

• Integer delay found via cross-correlation peak detection
• Fractional delay refined using parabolic interpolation or sinc fitting
• Accuracy must be within ±0.01 samples for effective DPD linearization
• Misalignment causes coefficient estimation bias and degraded ACLR

An ideal fractional delay filter implemented by shifting the sinc function by the desired fractional delay amount. Provides perfect reconstruction for bandlimited signals.

• Infinite impulse response in theory; requires windowing for practical FIR implementation
• Kaiser window commonly used to truncate while controlling sidelobe levels
• Optimal for wideband signals where Lagrange methods show passband droop
• Computationally more expensive than polynomial-based alternatives

An IIR filter approach that achieves fractional delay with unity magnitude response across all frequencies, ideal when amplitude distortion is unacceptable.

• Implemented using Thiran all-pass filters with closed-form coefficient formulas
• Provides exact unity gain; only phase response varies with frequency
• Group delay approximates the desired fractional delay at low frequencies
• Lower computational cost than equivalent FIR designs for narrowband applications

The application of fractional delay filters to synchronize the reference and feedback paths with sub-sample precision, critical for wideband and mmWave DPD systems.

• Sample-rate alignment alone insufficient when delay is not an integer multiple
• Fractional delay filter inserted in the reference path before coefficient extraction
• Misalignment of 0.1 samples can degrade ACLR by 3-5 dB in wideband scenarios
• Adaptive fractional delay estimation often combined with the DPD training loop