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Sample-rate conversion

Sample-rate conversion, also known as resampling or sampling-frequency conversion, is the process of changing the sampling rate of a discrete-time signal from an original rate F = 1/T to a new rate F' = 1/T', where T and T' are the respective sampling periods.^[1] This operation is fundamental in digital signal processing (DSP) and typically involves interpolation to increase the sampling rate or decimation to decrease it, often combined in rational ratios L/M (where L and M are integers) to achieve efficient conversion while preserving signal integrity.^[1] To prevent distortion from aliasing during decimation or imaging during interpolation, low-pass filtering is essential, ensuring the signal's frequency content remains within the Nyquist limits of the target rate.^[2] The core techniques for sample-rate conversion rely on multirate DSP structures, such as polyphase filters and multistage networks, which optimize computational efficiency by reducing the number of operations compared to single-stage implementations.^[1] For instance, in rational conversion by

[L/M](/page/L&M)

, the process inserts L-1 zeros between samples for upsampling, applies an anti-imaging low-pass filter, applies an anti-aliasing low-pass filter, and then retains every Mth sample (discarding the intervening samples) to avoid aliasing, with polyphase decomposition minimizing redundant computations.^[3] Finite impulse response (FIR) filters are commonly used for their linear phase properties, though infinite impulse response (IIR) filters can offer further efficiency in specific cases.^[4] These methods enable high-quality conversion even for irrational ratios through adaptive or arbitrary resampling algorithms.^[2] Sample-rate conversion plays a critical role in numerous applications, including digital audio processing for format compatibility (e.g., converting between 44.1 kHz and 48 kHz rates), telecommunications for channel rate adaptation, and image scaling in visual systems.^[4] In analog-to-digital (A/D) conversion and software-defined radio, it facilitates bandwidth-efficient signal handling by downsampling after anti-aliasing and upsampling for transmission, reducing hardware demands and power consumption.^[3] Its importance has grown with the proliferation of multirate systems in speech recognition, radar, and medical imaging, where precise rate adjustments enhance performance without excessive computational overhead.^[1]

Fundamentals

Definition and Motivation

Sample-rate conversion is the process of changing the sampling rate of a discrete-time signal to obtain a new discrete-time representation of the underlying continuous-time signal, while preserving as much of the original information as possible.^[5] This technique is fundamental in digital signal processing (DSP), where signals are represented as sequences of samples taken at regular intervals, and altering the rate allows adaptation to different system requirements without significant distortion.^[1] The primary motivation for sample-rate conversion stems from the need for compatibility across diverse digital systems that operate at varying sampling frequencies. For instance, consumer audio content recorded at 44.1 kHz for compact discs must often be converted to 48 kHz for professional broadcasting or video production workflows to ensure seamless integration.^[6] Additionally, it enables bandwidth efficiency by downsampling signals for storage or transmission over limited channels, reducing data volume while maintaining perceptual quality, and upsampling to match hardware constraints or enhance processing resolution in applications like telecommunications.^[4] These conversions are essential in multirate systems where mismatched rates could otherwise lead to inefficiencies or errors. Historically, sample-rate conversion emerged in the 1970s alongside the rise of digital signal processing, driven by the development of early digital audio and telecommunications standards that required handling multiple sampling rates. Pioneering work in this area, such as the foundational analyses of interpolation and decimation techniques, laid the groundwork for efficient multirate architectures in the late 1970s and early 1980s.^[1] At a high level, the process involves interpolation to increase the sampling rate by inserting new samples, or decimation to decrease it by selectively removing samples, invariably accompanied by low-pass filtering to mitigate aliasing during downsampling or imaging during upsampling.^[5] This aligns with the Nyquist-Shannon sampling theorem, which establishes the minimum rate needed to faithfully represent a signal's frequency content.^[1]

Nyquist-Shannon Theorem and Aliasing Risks

The Nyquist-Shannon sampling theorem establishes the fundamental limit for accurately capturing a continuous-time signal in the digital domain. It states that a bandlimited continuous-time signal with maximum frequency component B (in hertz) can be perfectly reconstructed from its uniformly spaced samples if the sampling frequency f_s satisfies f_s \geq 2B.^[7] This condition ensures that the discrete-time representation contains all necessary information to recover the original analog signal without loss, as derived from the theory of bandlimited functions and Fourier analysis. The threshold $2B represents the minimum sampling rate required, known as the Nyquist rate, beyond which higher frequencies cannot be distinguished from lower ones in the sampled sequence. A key consequence of violating this theorem is aliasing, a distortion where frequency components above the Nyquist frequency f_s/2—the highest frequency representable without overlap—fold back into the lower frequency band, masquerading as false lower-frequency signals. This phenomenon arises because sampling creates periodic replicas of the signal's spectrum centered at multiples of f_s, leading to overlap if the signal is not properly bandlimited. The aliased frequency f_\text{alias} for an original frequency f > f_s/2 is given by

f_\text{alias} = \left| f - k f_s \right|,

where k is the integer that minimizes the absolute value, typically mapping f_\text{alias} into the range [0, f_s/2). The Nyquist frequency f_s/2 thus defines the critical bandwidth: any signal energy exceeding this limit risks irreversible distortion upon sampling or processing.^[7] In the context of sample-rate conversion, these principles dictate necessary precautions to preserve signal integrity. Downsampling, which reduces the sampling rate, amplifies aliasing risks as the effective Nyquist frequency decreases, potentially causing high-frequency components to fold into the audible or relevant band; anti-aliasing low-pass filtering below the new f_s/2 is essential to attenuate such components prior to decimation. Conversely, upsampling increases the sampling rate and thereby raises the Nyquist frequency, avoiding the introduction of new aliased artifacts from existing signal content, though it may generate imaging artifacts—spectral replicas at higher frequencies—that require separate filtering to suppress. Adhering to the theorem ensures that rate changes maintain the signal's fidelity within its original bandwidth B.

Core Techniques

Upsampling

Upsampling increases the sampling rate of a discrete-time signal by an integer factor L, typically by inserting L-1 zero-valued samples between each original sample, which effectively multiplies the original sampling rate f_s by L. This process, known as expansion or zero-stuffing, prepares the signal for interpolation while avoiding the introduction of new information beyond the original bandwidth.^[8]^[9] For an input signal x, the upsampled signal y is generated such that the original samples are preserved at multiples of L, with zeros inserted elsewhere:

y = \begin{cases} x\left[\frac{m}{L}\right] & \text{if } m \mod L = 0 \\ 0 & \text{otherwise} \end{cases}

This operation compresses the spectrum of the original signal in the frequency domain, repeating it L times across the new Nyquist bandwidth L f_s / 2.^[9]^[10] The zero-insertion creates unwanted spectral images—replicas of the baseband spectrum centered at integer multiples of the original f_s—which can distort the signal if left unaddressed. To mitigate these imaging artifacts, an anti-imaging low-pass filter is applied immediately after upsampling, with its cutoff frequency set at the original Nyquist rate f_s / 2 to retain only the desired baseband while attenuating the images.^[11]^[12] In practice, upsampling is often used in audio processing to convert low-rate signals, such as telephone-quality audio at 8 kHz, to higher rates like 44.1 kHz for improved resolution in digital systems without exceeding the original frequency content.^[13]

Downsampling

Downsampling, also known as decimation, reduces the sampling rate of a discrete-time signal by an integer factor M > 1, effectively dividing the original rate f_s by M. This process involves applying a low-pass anti-aliasing filter to the input signal x to produce a filtered version z, followed by discarding M-1 out of every M samples to yield the output y = z[m M]. The filtering step is essential to prevent spectral aliasing that would otherwise distort the signal. The decimation operation can be expressed mathematically as

y = z[m M],

where z is the low-pass filtered version of x with a cutoff frequency of \pi / M radians per sample. In the frequency domain, unfiltered decimation compresses the spectrum by M and replicates it M times, causing high-frequency components to fold into the baseband as aliases. The anti-aliasing filter attenuates frequencies above the new Nyquist limit f_s / (2M), ensuring the output spectrum remains undistorted within |\omega| < \pi / M. Ideally, this filter has unit gain in the passband. Without the anti-aliasing filter, energy from frequencies exceeding \pi / M aliases into the lower band, potentially introducing audible artifacts or data loss in applications like audio processing. For instance, downsampling 48 kHz audio—typical for video production—to 8 kHz telephony standards (a factor of M = 6) requires filtering to attenuate components above 4 kHz, enabling bandwidth savings while preserving speech intelligibility. This technique, foundational to multirate signal processing, contrasts with upsampling by addressing aliasing rather than imaging, though the two operations are inverses in ideal bandlimited scenarios.

Rational-Factor Resampling

Rational-factor resampling refers to the process of converting a digital signal's sampling rate by a rational factor L/M, where L and M are coprime positive integers, resulting in a new sampling rate that is L/M times the original.^[1] This method integrates upsampling by the integer factor L and downsampling by the integer factor M to achieve arbitrary rational rate changes without requiring irrational computations.^[14] The process begins with upsampling, where L-1 zeros are inserted between each original sample to increase the rate by L, followed by lowpass filtering to suppress the spectral images introduced by zero insertion. This is then followed by downsampling, which involves lowpass filtering to prevent aliasing and decimation by retaining every M-th sample. The combined lowpass filter for the overall system has a cutoff frequency of \min(\pi/L, \pi/M) in the normalized frequency scale at the intermediate sampling rate, corresponding physically to \min(f_s/2, f_s'/2), where f_s is the original sampling rate and f_s' is the new rate.^[1] The output signal y is thus obtained via bandlimited interpolation, approximated as

y \approx \sum_k x \, h\left( \frac{n L - k M}{M} \right),

where x are the input samples and h(\cdot) is the impulse response of the ideal lowpass filter designed for the rational conversion.^[14] Efficiency in rational-factor resampling stems from implementations that compute only the necessary output samples directly, bypassing the storage and processing of the full intermediate signal at rate L f_s, which would otherwise multiply computational demands by L. For example, converting audio from 44.1 kHz to 48 kHz employs L=160 and M=147, as $44.1 \times 160 = 48 \times 147 = 7056 Hz in a common multiple, enabling exact rational conversion with reduced operations compared to naive upsampling to an excessively high rate.^[1] A key challenge arises when the desired rate ratio is irrational, such as the exact $44.1/48 \approx 0.91875, necessitating a close rational approximation like $147/160 to minimize distortion; the approximation error decreases with higher L and M, but increases filter complexity and computation.^[14]

Advanced Algorithms

Interpolation Methods

Interpolation methods are essential for estimating intermediate sample values when increasing the sampling rate in sample-rate conversion, enabling the reconstruction of a continuous-time signal from discrete samples before resampling at the higher rate. These methods vary in complexity and accuracy, balancing computational demands with the fidelity of the reconstructed signal. The choice depends on the application, with simpler techniques suiting real-time constraints and more advanced ones prioritizing quality in offline processing.^[1] The simplest approach is nearest-neighbor interpolation, also known as zero-order hold, where each new sample is assigned the value of the closest original sample. Mathematically, for an output sample index n, the time position t = n \cdot (f_s^{\text{new}} / f_s^{\text{old}}) is computed, and the output is y = x[\round(t)], with x[\cdot] denoting the input samples. This method incurs zero computational cost beyond the ratio calculation, making it ideal for resource-limited systems. However, it suffers from high aliasing and severe waveform distortion due to the lack of smoothing between samples.^[15]^[16] A step up in sophistication is linear interpolation, which connects adjacent original samples with straight lines to estimate new values. The formula is y = x[\floor(t)] + (t - \floor(t)) \cdot (x[\ceil(t)] - x[\floor(t)]), where \floor(\cdot) and \ceil(\cdot) are the floor and ceiling functions, respectively. This requires only a few arithmetic operations per sample, offering low complexity suitable for many embedded applications. While it provides smoother transitions than nearest-neighbor, linear interpolation introduces phase distortion and attenuates higher frequencies, leading to reduced fidelity for bandlimited signals.^[16]^[1] The theoretically ideal method is sinc interpolation, derived from the Nyquist-Shannon sampling theorem, which enables perfect reconstruction of a bandlimited signal. The continuous-time reconstruction is given by

y(t) = \sum_{k=-\infty}^{\infty} x \cdot \sinc\left( \frac{t - k T_s}{T_s} \right),

where \sinc(u) = \sin(\pi u)/(\pi u) is the normalized sinc function and T_s = 1/f_s^{\text{old}} is the original sampling period. Discrete samples at the new rate are obtained by evaluating this at the corresponding times. This approach eliminates aliasing and distortion for signals below the Nyquist frequency but demands infinite computation due to the sinc function's infinite extent, rendering it impractical without truncation and windowing approximations.^[1] In practice, these methods trade off computational complexity against reconstruction fidelity. Nearest-neighbor offers negligible overhead but poor quality with prominent aliasing artifacts, while linear interpolation improves smoothness at modest cost yet compromises on spectral accuracy. Sinc interpolation sets the fidelity benchmark, serving as the basis for practical finite impulse response (FIR) filters that approximate its ideal response through truncation, though at significantly higher complexity. Seminal analyses highlight that optimal designs favor sinc-based approximations for high-quality applications, such as professional audio, where aliasing must be minimized.^[1]

Polyphase Filter Structures

Polyphase filter structures exploit the mathematical properties of multirate systems to implement sample-rate conversion with significantly reduced computational complexity. In polyphase decomposition, a prototype filter impulse response h is partitioned into L sub-filters (for upsampling by integer factor L) or M sub-filters (for downsampling by integer factor M), where each sub-filter operates at the original sampling rate. This breakdown transforms the full-rate filtering operation into parallel branches, each handling a decimated version of the input signal. The z-transform representation of the filter is expressed as H(z) = \sum_{k=0}^{L-1} z^{-k} E_k(z^L), where E_k(z) are the polyphase components given by E_k(z) = \sum_{n} h[nL + k] z^{-n}, which rearranges into contributions from the polyphase branches, enabling efficient computation without explicitly inserting zeros.^[17] A key enabler of this efficiency is the noble identities, which permit commuting the filter with the rate-change operator under certain conditions, such as when the filter is expressed in polyphase form. For upsampling, the identity allows the polyphase sub-filters to precede the upsampler, performing operations at the lower input rate; similarly, for downsampling, sub-filters follow the downsampler. This commutativity ensures that filtering occurs at the lower sampling rate, avoiding unnecessary computations on zero-valued samples in interpolation or redundant processing before decimation. The resulting structure reduces complexity from O(N) operations per output sample in direct convolution (where N is the filter length) to approximately O(N/L), achieving a speedup factor of roughly L while maintaining the same frequency response. Polyphase implementations are particularly advantageous for long filters, as they scale linearly with filter length but benefit multiplicatively from the rate factor.^[18]^[19] For rational resampling by factor L/M, where L and M are coprime integers, the polyphase structure combines an interpolator followed by a decimator into a single time-multiplexed architecture using a commutator. The input signal is fed into \max(L, M) polyphase branches of the anti-imaging/anti-aliasing filter, with a commutator selecting outputs at the desired rate, effectively interleaving the sub-filter responses. This unified design minimizes intermediate sample rates and storage, making it suitable for hardware-constrained environments. For instance, in real-time audio sample-rate conversion, a polyphase sinc filter—derived from the ideal low-pass prototype—reduces multiplications by a factor approximately equal to L or M, enabling low-latency processing for conversions like 44.1 kHz to 48 kHz without perceptible artifacts.^[18]^[20] Modern variants extend polyphase structures to adaptive scenarios, particularly in software-defined radio (SDR), where variable or time-varying rates are common. Adaptive polyphase filters dynamically select or interpolate filter phases based on the instantaneous rate ratio, using a fractional delay mechanism to handle non-integer shifts. This approach supports seamless rate adjustments for diverse modulation schemes and channel conditions, with complexity remaining proportional to the filter length rather than the rate variation. Such implementations have been demonstrated to achieve high efficiency in SDR terminals, balancing quality and resource use for applications like multi-standard wireless communication.^[21]

Applications

Audio Systems

In audio systems, sample-rate conversion is essential for compatibility across diverse standards and devices. The compact disc (CD) format employs a sample rate of 44.1 kHz, while professional audio and DVD audio typically use 48 kHz, and high-resolution audio often utilizes 96 kHz to capture extended frequency ranges. A common conversion in mixing workflows involves resampling from 44.1 kHz to 48 kHz to align consumer and professional formats during post-production.^[22] Digital audio workstations (DAWs) frequently apply sample-rate conversion during export, often combined with dithering to minimize quantization noise when reducing bit depth alongside rate changes.^[23] Streaming services perform conversions to adapt high-resolution source material to device-specific playback rates, ensuring seamless delivery across varied hardware.^[6] In vinyl-to-digital transfers, analog signals are digitized at rates like 48 kHz or higher, with subsequent conversion to standard rates such as 44.1 kHz for archiving or distribution.^[24] A key application is sample-rate conversion during MP3 encoding, where lowering the rate from 48 kHz to 44.1 kHz can help reduce bitrate demands while preserving perceptual quality through efficient compression.^[25] Asynchronous sample-rate conversion (ASRC) addresses clock drift in playback devices, dynamically adjusting rates between mismatched clocks in sources and receivers to prevent buffer overflows or underruns.^[26] The Audio Engineering Society (AES) provides guidelines emphasizing 48 kHz as a preferred rate for professional interchange to limit distortion in audio chains, recommending high-quality converters that maintain signal integrity during rate changes.^[27] Rational resampling techniques are commonly employed for these non-integer rate ratios in audio systems.^[6]

Video and Multimedia

Sample-rate conversion in video and multimedia involves adjusting frame rates and pixel rates to accommodate diverse standards across film, broadcast television, and digital platforms. Traditional film is captured at 24 frames per second (fps), while broadcast standards vary: NTSC regions use approximately 29.97 or 59.94 fps for interlaced or progressive video, PAL regions employ 25 or 50 fps, and high-definition television (HDTV) often requires conversions such as from 24 fps to 60 fps to ensure compatibility with progressive displays. These adjustments prevent temporal artifacts and maintain visual fluidity during distribution. Temporal resampling addresses frame rate changes by interpolating or decimating frames to match target rates, while spatial resampling handles pixel rate adjustments during video resizing, such as scaling resolutions from standard-definition to high-definition formats. Motion-compensated interpolation enhances these processes by estimating object motion across frames to generate intermediate ones, reducing judder in conversions like pulldown sequences.^[28] For instance, in slow-motion effects, upsampling via frame interpolation inserts additional frames to extend playback duration without introducing artifacts such as judder. A prominent example is the 3:2 pulldown technique, which converts 24 fps film to 29.97 fps NTSC video by repeating fields in a 3:2 pattern over five fields, ensuring smooth transfer while preserving motion integrity in film-to-video workflows. In digital video codecs like H.264/AVC, sample-rate conversion supports adaptive streaming by enabling frame rate adjustments during encoding, allowing content to adapt to varying bandwidths and playback devices while maintaining synchronization. Multimedia integration, such as in Blu-ray authoring, demands simultaneous sample-rate conversion for audio and video to uphold lip-sync, where video frame rates (e.g., 24 fps) must align with audio sampling rates (e.g., 48 kHz) through precise temporal processing to avoid drift in fractional-rate environments.^[29] This ensures seamless playback across hybrid media, with standards emphasizing minimal latency in conversion to preserve perceptual quality.^[30]

Performance Considerations

Artifacts and Quality Metrics

Sample-rate conversion can introduce various artifacts that degrade the fidelity of the reconstructed signal, primarily due to imperfect filtering or approximation of the ideal reconstruction process. Aliasing manifests as unwanted "ghost frequencies" that fold into the baseband when downsampling without adequate low-pass filtering, violating the Nyquist criterion and creating audible or visible distortions in audio and video signals. Imaging occurs during upsampling as high-frequency echoes or replicas of the original spectrum appear above the new Nyquist frequency, often resulting from insufficient anti-imaging filters that fail to attenuate these spectral images. Jitter arises from poor filtering implementations, introducing timing irregularities that manifest as phase noise or modulation artifacts, particularly in real-time systems where filter delays vary. Phase distortion is common in linear-phase interpolation methods, where group delay variations across frequencies lead to temporal smearing, especially noticeable in transient signals like percussive audio. To quantify the quality of sample-rate conversion, several metrics are employed to assess deviations from an ideal reconstruction, often using sinc interpolation as a reference benchmark that theoretically minimizes such artifacts. The signal-to-noise ratio (SNR) measures the power ratio of the desired signal to the noise introduced by conversion errors, calculated as:

\text{SNR} = 10 \log_{10} \left( \frac{P_{\text{signal}}}{P_{\text{noise}}} \right)

where P_{\text{noise}} encompasses quantization, aliasing, and imaging contributions post-conversion. Mean squared error (MSE) evaluates the average squared difference between the converted signal and an ideal bandlimited reconstruction, providing a simple objective measure of overall distortion. For audio applications, perceptual evaluation models such as PEAQ (Perceptual Evaluation of Audio Quality) incorporate human auditory models to predict subjective quality, accounting for masking effects and frequency selectivity beyond raw error metrics. Evaluation of conversion quality often emphasizes frequency-domain characteristics, with high-quality systems requiring a flat frequency response and minimal passband ripple, typically less than 0.1 dB, to preserve spectral integrity without introducing coloration or attenuation variations. These metrics collectively ensure that artifacts remain below perceptible thresholds, with SNR values exceeding 90 dB considered professional-grade for critical listening environments.

Optimization and Hardware Implementation

Software implementations of sample-rate conversion (SRC) prioritize computational efficiency and audio fidelity, with libraries such as libsamplerate, also known as Secret Rabbit Code, providing high-quality conversion for arbitrary and time-varying ratios using polyphase filtering techniques.^[31] Developed by Erik de Castro Lopo, this open-source library supports multiple conversion qualities, from linear interpolation for low CPU usage to sinc-based methods for near-theoretical performance, making it suitable for real-time audio processing in applications like music production software.^[31] For handling arbitrary ratios beyond rational factors, FFT-based methods offer a frequency-domain approach that resamples signals by modifying the spectral content of a large discrete Fourier transform buffer before inverse transformation.^[32] These "giant FFT" techniques, as detailed in works by Bilbao and Parker, enable efficient non-integer conversions with reduced aliasing through phase-adjusted spectral interpolation, achieving up to 10-20 times faster processing than time-domain equivalents for high-resolution audio.^[32] Such methods are integrated into libraries like SoX and FFmpeg for batch processing of multimedia files.^[33] In hardware, asynchronous sample-rate conversion (ASRC) chips are widely used in digital-to-analog converters (DACs) to match disparate input and output clock rates in real-time, preventing buffer overflows or underruns in systems like professional audio interfaces.^[34] Devices such as the Cirrus Logic CS8420 employ polyphase FIR filters to achieve this synchronization with minimal jitter, supporting input rates from 8 kHz to 108 kHz while maintaining signal integrity.^[35] Similarly, FPGA-based implementations leverage polyphase structures for low-latency SRC, as seen in Intona's IP cores, which utilize reconfigurable logic to process up to 230 kHz audio with under 1 ms delay and reduced resource utilization compared to software equivalents.^[36] Optimizations for dynamic environments include variable-rate SRC algorithms that adapt to fluctuating input rates, essential for adaptive streaming in network audio systems where bandwidth varies.^[37] The XMOS SRC library, for instance, supports asynchronous modes that track clock drift in real-time using phase-locked loops, enabling seamless playback of variable-rate sources like internet radio without audible glitches.^[37] Hybrid analog-digital approaches further enhance performance by combining digital resampling with analog anti-aliasing filters in DAC pipelines, minimizing distortion in high-fidelity playback; Benchmark Media's DAC2 exemplifies this by integrating post-digital analog processing to achieve total harmonic distortion below -120 dB.^[38] As of 2025, recent advances incorporate AI-assisted filter design to optimize SRC for neural audio synthesis, where multirate processing is critical for sample-rate-independent recurrent neural networks (RNNs).^[39] Research by Carson et al. demonstrates two-stage resampling filters—combining half-band IIR and Kaiser-window FIR designs—trained via neural optimization to reduce computational overhead in audio effect RNNs, enabling real-time synthesis at varying rates with up to 30% lower aliasing in generated signals.^[39] These methods, published in IEEE Transactions on Audio, Speech, and Language Processing, facilitate efficient integration of SRC in AI-driven tools for music generation and processing.^[40]

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[41]
Resampling filter design for Multirate Neural Audio Effect Processing
Jun 3, 2025 · We investigate several resampling filter designs and show that a two-stage design consisting of a half-band IIR filter cascaded with a Kaiser ...Missing: assisted conversion 2024