# Multirate DSP, part 3: ADC oversampling

*Order this book today at
www.elsevierdirect.com or by calling 1-800-545-2522 and receive an additional 20% discount. Use promotion code 92562 when ordering. Offer expires 06/30/2008. Valid only in North America.
*

*Part 2 show how to change the sampling rate by a non-integer factor. It also looks at multistage decimation and polyphase filters. Part 4 shows how to undersample bandpass signals.*

**12.3 Oversampling of Analog-to-Digital Conversion**

Oversampling of the analog signal has become popular in DSP industry to improve resolution of analog-to-digital conversion (ADC). Oversampling uses a sampling rate, which is much higher than the Nyquist rate. We can define an oversampling ratio as

*f*_{s}/(2*f*_{max}) >> 1. (12.14)

The benefits from an oversampling ADC include:

- helping to design a simple analog anti-aliasing filter before ADC, and
- reducing the ADC noise floor with possible noise shaping so that a low-resolution ADC can be used.

**12.3.1 Oversampling and Analog-to-Digital Conversion Resolution**

To begin with developing the relation between oversampling and ADC resolution, we first summarize the regular ADC and some useful definitions discussed in Chapter 2:

Quantization noise power = σ_{q}^{2}= Δ^{2}/12 (12.15)

Quantization step = Δ = *A*/(2* ^{n}*) (12.16)

where

*A *= full range of the analog signal to be digitized

*n *= number of bits per sample (ADC resolution).

Substituting Equation (12.16) into Equation (12.15), we have:

Quantization noise power = σ_{q}^{2}= (A^{2}/12) × 2^{-2n}. (12.17)

The power spectral density of the quantization noise with an assumption of uniform probability distribution is shown in Figure 12-23. Note that this assumption is true for quantizing a uniformly distributed signal in a full range with a sufficiently long duration. It is not generally true in practice. See research papers by Lipshitz et al. (1992) and Maher (1992). However, using the assumption will guide us for some useful results for oversampling systems.

*Figure 12-23. Regular ADC system.*

The quantization noise power is the area obtained from integrating the power spectral density function in the range of –*f*_{s}/2 to *f*_{s}/2. Now let us examine the oversampling ADC, where the sampling rate is much bigger than that of the regular ADC; that is *f*_{s }>> 2*f*_{max}. The scheme is shown in Figure 12-24.

*Figure 12-24. Oversampling ADC system.*

As we can see, oversampling can reduce the level of noise power spectral density. After the decimation process with the decimation filter, only a portion of quantization noise power in the range from –*f*_{max }to *f*_{max }is kept in the DSP system. We call this an *in-band frequency range*. In Figure 12-24, the shaded area, which is the quantization noise power, is given by

Assuming that the regular ADC shown in Figure 12-23 and the oversampling ADC shown in Figure 12-24 are equivalent, we set their quantization noise powers to be the same to obtain

(*A*^{2}/12) × 2^{–2n} = ((2*f*_{max})/*f*_{s}) × (*A*^{2}/12) × 2^{–2m}. (12.19)

Equation (12.19) leads to two useful equations for applications:

*n *= *m *+ 0.5 × log_{2} (*f*_{s}/(2*f*_{max})) (12.20) and

*f*_{s }= 2*f*_{max }× 2^{2(n – m)}, (12.21)

where

*f*_{s} = sampling rate in the oversampling DSP system

*f*_{max }= maximum frequency of the analog signal

*m *= number of bits per sample in the oversampling DSP system

*n *= number of bits per sample in the regular DSP system using the minimum sampling rate

From Equation (12.20) and given the number of bits (*m*) used in the oversampling scheme, we can determine the number of bits per sample equivalent to the regular ADC. On the other hand, given the number of bits in the oversampling ADC, we can determine the required oversampling rate so that the oversampling ADC is equivalent to the regular ADC with the larger number of bits per sample (*n*). Let us look at the following examples.

**Example 12.7.**

Given an oversampling audio DSP system with the following attributes, determine the oversampling rate to improve the ADC to 16-bit resolution:

- Maximum audio input frequency of 20 kHz and
- ADC resolution of 14 bits,

Solution: Based on the specifications, we have

*f*_{max }= 20 kHz, *m *= 14 bits, and *n *= 16 bits.

Using Equation (12.21) leads to

*f*_{s }= 2*f*_{max } × 2^{2(n – m)}_{ }= 2 × 20 × 2^{2(16 – 14)}_{ }= 640 kHz.

Since *f*_{s}/(2*f*_{max}) = 2^{4}, we see that each doubling of the minimum sampling rate (2*f*_{max }= 40 kHz) will increase the resolution by a half bit.

**Example 12.8.**

Given an oversampling audio DSP system with the following attributes, determine the equivalent ADC resolution:

- Maximum audio input frequency = 4 kHz
- ADC resolution = 8 bits
- Sampling rate = 80 MHz.

Solution: Since *f*_{max }= 4 kHz, *f*_{s }= 80 kHz, and *m *= 8 bits, applying Equation (12.20) yields

*n *= *m *+ 0.5 × log_{2} (*f*_{s}/(2*f*_{max})) = 8 + 0.5 × log_{2} (80000 *kHz*/(2 × 4 *kHz*)) ≈ 15 bits.