Modeling the Quantization Staircase Function

Salman Aslam, Aaron Bobick, Christopher Barnes,Georgia Institute of Technology

(a) Integrating a square wave with 50%duty cycle to yield a right angled isoscelestriangle.

(b) Computing y.

(c) Computing the output, x + z

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−10

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Model 1Model 2

(d) Final result.

Figure 1: Modeling a uniform midriser quantizer function.

An analytical expression for the uniform midrise quantizer function can be written as a sum ofthe input and a residual sawtooth wave. We propose an equivalent but alternate approach. Althoughit is well known that no clever mathematical manipulation can yield extra information, nevertheless,an alternate form of expression can and has on several occassions proven useful. Good sources ofinformation on quantizers are [1] and [2].

For a given quantizer input x, our goal is to find the quantizer output, x+z, as given in Figure 1.We integrate a square wave Figure 1(a) to yield a triangle wave with discrete fourier coefficients,ct[k] which are linear in the quantization parameter Qp. We rotate this triangle wave 45 ◦ counterclockwise (Figure 1(b)) to find y, which in turn is used to find the ouput x + z (Figure 1(c)).

y[x] = Qp

[1√8− 0.2866 cos(

2πx

Qp) − 0.0318 cos(

6πx

Qp) − 0.0114 cos(

10πx

Qp)]

(1)

z can take on two different values, z1 and z2, depending on the input, and this is modeled usinganother square wave. In Figure 1(b), z = z1 if x is between points A and D and z = z2 if x isbetween points D and C. z1 and z2 are given by z1 = 0.707y1 and z2 = 0.707(1.414Qp)− y2. x + zis plotted in Figure 1(d).

[1] A. Gersho and R. M. Gray, Vector Quantization and Signal Compression (The Springer Inter-national Series in Engineering and Computer Science). Springer, 1991.[2] N. Jayant, Signal Compression: Coding of Speech, Audio, Text, Image and Video (Selected Topicsin Electronics and Systems , Vol 9). World Scientific Pub Co Inc, 1997.

2010 Data Compression Conference

1068-0314/10 $26.00 © 2010 IEEE

DOI 10.1109/DCC.2010.89

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