Interpolation (Upsampling)

Increasing the Data Rate

The previous lesson covered decimation — reducing the sample rate by keeping every M-th sample. Interpolation is the dual operation: it increases the sample rate by a factor of L, producing L output samples for every input sample. Where decimation discards samples, interpolation creates new ones.

The need to increase a sample rate arises constantly in practice. A digital-to-analog converter (DAC) operates most accurately when fed at a high rate; oversampling the audio stream by 128× before conversion dramatically reduces the requirements on the analog reconstruction filter. An OFDM transmitter must insert pilot tones and guard intervals, which implicitly changes the effective sample rate. In all of these cases, interpolation by integer factor L provides the foundation.

Step 1 — Zero Insertion

The first stage of an interpolator is deceptively simple: insert L−1 zeros between every pair of consecutive input samples. If the input sequence is x[n], the zero-inserted sequence v[m] is defined as:

Why zeros and not repeated samples? Repeating the same sample L times is equivalent to multiplying by a rectangular pulse, which introduces severe distortion. Inserting zeros is mathematically cleaner: the z-transform of v[m] is simply V(z) = X(z^L), a frequency-compressed version of the original spectrum with no distortion — only extra spectral copies (images) that the subsequent filter will remove.

The Imaging Problem

Zero insertion compresses the spectrum by a factor of L along the frequency axis, and simultaneously creates L−1 spectral replicas of the original signal — called images — evenly spaced at multiples of 2π/L. These images are mathematical artifacts, not real signal content, but they would appear as tones or distortion in the output if left unfiltered.

Consider upsampling by L = 4 a signal with bandwidth B. After zero insertion at the new rate 4f_s, the spectrum contains the original band plus three images at shifts of π/2, π, and 3π/2 radians. The anti-imaging filter erases these three copies, leaving only the band near DC — the desired upsampled signal.

Step 2 — The Anti-imaging Filter

The second stage is a lowpass filter with cutoff frequency ω_c = π/L (evaluated at the higher output sample rate). This filter passes the main spectral copy centered at DC and suppresses all L−1 images. The gain of the filter in the passband is set to L to compensate for the factor-of-L amplitude reduction caused by inserting zeros.

As with decimation, FIR filters are the standard choice for interpolation. They are linear-phase, stable, and easily designed using the windowed-sinc or Parks-McClellan methods. The filter order required for a given stopband attenuation depends on the transition bandwidth — a small transition band (i.e., a signal occupying most of π/L) demands more taps.

What the Filter Does in the Frequency Domain

From a frequency-domain perspective, interpolation is elegant. The zero-insertion stage compresses the spectrum and tiles it L times across [0, 2π]. The lowpass filter then selects only the central tile — the one centered at DC — and scales it up by L. The result is a signal whose spectrum is an amplitude-scaled version of the original, now occupying [−π/L, +π/L] within the wider [−π, +π] band of the output.

In the time domain, the filter effectively computes a weighted interpolation between the known input samples. The impulse response h[n] = L · sinc(n/L) is the ideal interpolating kernel — a windowed version of which is used in practice. Each output sample is a linear combination of nearby input samples, with sinc-shaped weights that peak at the true sample locations and taper smoothly between them.

Efficient Implementation: Polyphase Decomposition

A naive interpolator computes the full anti-imaging filter at the high output rate Lf_s, even though L−1 out of every L input samples are zero — wasted multiplications. The polyphase decomposition reorganizes the filter to operate at the lower input rate.

The anti-imaging FIR filter h[n] of length N is split into L polyphase branches E_k[n] = h[nL + k] for k = 0, 1, …, L−1. Each branch has approximately N/L taps and processes the input sequence x[n] at the original rate f_s. The L branch outputs are commutated (interleaved) to form the output stream at the higher rate Lf_s.

The polyphase interpolator is the standard structure in audio sample-rate converters, DAC oversampling chains, and software-defined radio transmitters. It avoids all multiplications with zero-valued input samples and scales the filter cost proportional to N/L rather than N.

Practical Applications

DAC oversampling: Audio DACs routinely upsample by 128× or 256× before digital-to-analog conversion. Operating the DAC at a very high rate pushes the analog anti-aliasing filter requirements out of the audible band, where a gentle rolloff suffices — greatly simplifying the analog circuitry and reducing distortion.

Pulse shaping in digital communications: In systems such as 4G LTE or DVB, the baseband symbol stream is upsampled and filtered with a root-raised cosine (RRC) filter. The upsampling step inserts the necessary samples between symbols so the RRC filter can shape each pulse to minimize inter-symbol interference.

Image and video processing: Zooming and super-resolution both require interpolation: new pixel values must be synthesized between the sampled grid points of the original image. The anti-imaging filter is the spatial equivalent of the temporal anti-imaging filter in audio — it prevents ringing and aliasing in the upscaled image.