Filling the Silence Between Two Good Samples

The Teac A-3300SX came with two reels from the previous owner’s estate. One is unlabelled. The other says “DAD’S BIRTHDAY 1983” in faded marker. I haven’t played either — the heads need demagnetizing and the pinch roller has gone hard — but I’ve been reading about what happens when I finally do.

Tape sheds. Forty-three years of oxide binder slowly releasing its grip means microscopic gaps in the magnetic signal. When a digital transfer hits one of these dropouts, the waveform just… vanishes for a few samples. The restoration software has to guess.

The simplest guess is a straight line. If sample 1000 reads 0.42 and sample 1002 reads 0.58, then sample 1001 was probably somewhere around 0.50. This is linear interpolation: given two known points, assume the unknown point lies on the line between them.

func interpolate(samples: [Double], missingAt i: Int) -> Double {
    guard i > 0, i < samples.count - 1 else { return 0.0 }
    let before = samples[i - 1]
    let after = samples[i + 1]
    return (before + after) / 2.0
}

var audio: [Double] = [0.3, 0.42, .nan, 0.58, 0.61]
audio[2] = interpolate(samples: audio, missingAt: 2)
print(audio)  // [0.3, 0.42, 0.5, 0.58, 0.61]

sub interpolate {
    my ($samples, $i) = @_;
    return ($samples->[$i-1] + $samples->[$i+1]) / 2;
}

my @audio = (0.3, 0.42, undef, 0.58, 0.61);
$audio[2] = interpolate(\@audio, 2);
print join(", ", @audio), "\n";  # 0.3, 0.42, 0.5, 0.58, 0.61

The formula is ancient — Babylonian astronomers used it to predict planetary positions between observations. By the 1970s, it had become the foundation of digital audio’s dropout compensation. When Denon and Sony built the first commercial PCM recorders, they included interpolation circuits to paper over the inevitable tape errors.

It works remarkably well for short gaps. A single missing sample at 44.1 kHz represents 22.7 microseconds of audio — a span so brief that even a straight line sounds natural. The ear forgives small lies.

Longer gaps need higher-order interpolation: cubic splines, polynomial fits, spectral analysis. But for the routine oxide shed of a well-stored tape, linear interpolation handles most of it. The math is trivial. The restoration it enables is not.

I still haven’t pressed play on those reels. The machine isn’t ready. Neither, I think, am I.