2:1 downsampling

Figure 21.5 An analog filter for 2×-oversampling is much less demanding than a filter for direct sampling, because the difficult part of filtering – achieving a response comparable to that of Figure 21.4 – is relegated to the digital domain.

0

1

For an explanation of transition ratio, see page 212.

inexpensive. These circumstances have led to the emergence of oversampling as an economical alternative to complex analog presampling (“antialiasing”) and postsampling (reconstruction) filters.

The characteristics of a conventional analog presampling filter are critical: Attenuation must be quite low up to about 0.4 times the sample rate, and quite high above that. In a presampling filter for studio video, attenuation must be less than 1 dB or so up to about 5.5 MHz, and better than 40 or 50 dB above 6.75 MHz. This is a demanding transition ratio ∆ω /ω S. Figure 21.4 above (top) sketches the filter template of a conventional analog presampling filter.

An oversampling A-to-D converter operates at a multiple of the ultimate sampling rate – say at 27 MHz, twice the rate of BT.601 video. The converter is preceded by a cheap analog filter that severely attenuates components at 13.5 MHz and above. However, its characteristics between 5.5 MHz and 13.5 MHz are not critical. The demanding aspects of filtering in that region are left to a digital 2:1 downsampler. The transition ratio ∆ω /ω S of the analog filter is greatly relaxed compared to direct conversion. In today’s technology, the cost of the digital downsampler is less than the difference in cost between excellent and mediocre

In certain FIR filters whose corner is exactly 0.25fS, half the coefficients are zero. This leads to a considerable reduction in complexity.

In the common case of interpolation horizontally across an image row, the argument x is horizontal position. Interpolating along the time axis, as in digital audio sample rate conversion, you could use the symbol t to represent time.

In computer graphics, the linear interpolation operation is often called LIRP (pronounced lerp).

analog filtering. Complexity is moved from the analog domain to the digital domain; total system cost is reduced. Figure 21.5 (on page 225) sketches the template of an analog presampling filter appropriate for use preceding a 2x oversampled A-to-D converter.

Figure 20.25, on page 215, showed the response of a 55-tap filter having a corner frequency of 0.25fS. This is a halfband filter, intended for use following

a 2×-oversampled A-to-D converter.

The approach to 2×-oversampled D-to-A conversion is comparable. The D-to-A device operates at 27 MHz; it is presented with a datastream that has been upsampled by a 1:2 ratio. For each input sample, the 2×-oversampling filter computes 2 output samples. One is computed at the effective location of the input sample, and the other is computed at an effective location halfway between input samples. The filter attenuates power between 6.75 MHz and 13.5 MHz. The analog postsampling filter need only reject components at and above 13.5 MHz. As in the 2×-oversampling A-to-D conversion, its performance between 6.75 MHz and 13.5 MHz isn’t critical.

Interpolation

In mathematics, interpolation is the process of computing the value of a function or a putative function (call it g˜ ), for an arbitrary argument (x), given several function argument and value pairs [xi, si]. There are many methods for interpolating, and many methods for constructing functions that interpolate.

Given two sample pairs [x0, s0] and [x1, s1], the linear interpolation function has this form:

I symbolize the interpolating function as g˜ ; the symbol f is already taken to represent frequency. I write g with a tilde (g˜ ) to emphasize that it is an approximation.

The linear interpolation function can be rewritten as a weighted sum of the neighboring samples s0 and s1:

Julius O. Smith calls this WaringLagrange interpolation, since Waring published it 16 years before Lagrange. See Smith’s Digital Audio Resampling Home Page, <wwwccrma.stanford.edu/~jos/resample>.

The weights depend upon the x (or t) coordinate:

Lagrange interpolation

J.L. Lagrange (1736–1813) developed a method of interpolation using polynomials. A cubic interpolation function is a polynomial of this form:

Interpolation involves choosing appropriate coefficients a, b, c, and d, based upon the given argument/value pairs [xj, sj]. Lagrange described a simple and elegant way of computing the coefficients.

Linear interpolation is just a special case of Lagrange interpolation of the first degree. (Directly using the value of the nearest neighbor can be considered zeroorder interpolation.) There is a second-degree (quadratic) form; it is rarely used in signal processing.

In mathematics, to interpolate refers to the process that I have described. However, the same word is used to denote the property whereby an interpolating function produces values exactly equal to the original sample values (si) at the original sample coordinates (xi). The Lagrange functions exhibit this property. You might guess that this property is a requirement of any interpolating function. However, in signal processing this is not a requirement – in fact, the interpolation functions used in video and audio rarely pass exactly through the original sample values. As a consequence of using the terminology of mathematics, in video we have the seemingly paradoxical situation that interpolation functions usually do not “interpolate”!

In principle, cubic interpolation could be undertaken for any argument x, even values outside the x-coordi- nate range of the four input samples. (Evaluation outside the interval [x-1, x2] would be called extrapolation.) In digital video and audio, we limit x to the range between x0 and x1, so as to estimate the signal in the interval between the central two samples. To evaluate outside this interval, we substitute the input sample values [s-1, s0, s1, s2] appropriately – for example, to

Figure 21.6 Cubic interpolation of a signal starts with equally spaced samples, in this example 47, 42, 43, and 46. The underlying function is estimated to be a cubic polynomial that passes through (“interpolates”) all four samples. The polynomial is evaluated between the two central samples, as shown by the black segment. Here, evaluation is at phase offset . If the underlying function isn’t a polynomial, small errors are produced.

Sample coordinate

Unser, Michael (1999), “Splines: A perfect fit for signal and image processing,” IEEE Signal Processing Magazine: 22–38 (Nov.).

evaluate between s1 and s2, we shift the input sample values left one place.

With uniform sampling (as in conventional digital video), when interpolating between the two central samples the argument x can be recast as the phase offset, or the fractional phase ( , phi), at which a new sample is required between two central samples. (See Equation 21.5.) In abstract terms, lies between 0 and 1; in hardware, it is implemented as a binary or

a rational fraction. In video, a 1-D interpolator is usually an FIR filter whose coefficients are functions of the phase offset. The weighting coefficients (ci) are functions of the phase offset; they can be considered as basis functions.

In signal processing, cubic (third-degree) interpolation is often used; the situation is sketched in

Figure 21.6 above. In linear interpolation, one neighbor to the left and one to the right are needed. In cubic interpolation, we ordinarily interpolate in the central interval, using two original samples to the left and two to the right of the desired sample instant.

Equation 21.2 can be reformulated:

g( ) = c−1( ) s−1 + c0( ) s0 + c1( ) s1 + c2( ) s2 Eq 21.6

The function takes four sample values [s-1, s0, s1, s2] surrounding the interval of interest, and the phase offset between 0 and 1. The coefficients (ci) are now functions of the argument ; the interpolator forms