Multi-Channel Digital Conversion of Audio Sampling Rates

	ABOUT THE AUTHORS Adam Dabrowski received an M.Sc. degree in electrical engineering from Poznan University of Technology in 1976 and an M.Sc. degree in mathematics from Adam Mickiewicz University in 1977. In 1983 he received a Ph.D. degree in electronics. He is now a full professor at the Institute of Electronics and Telecommunication and the head of the Division for Signal Processing and Electronic Systems at Poznan University of Technology. More...

Audio signals are currently represented digitally with various sampling rates: 8, 16, 22.05, 24, 32, 44.1, 48, and 96 ksamples/s. In order to achieve high spatial realism of audio signals, you should use a multi-channel representation. Two typical systems are stereo sound (two front channels) and 5.1 surround sound (three front channels, two rear channels, and a subwoofer). An online multi-channel sampling rate conversion is a very important task that you can realize using modern DSPs, which feature a host port interface (HPI) and a buffered serial port (BSP). Figure 1 presents an experimental DSP hardware system for multi-channel sampling-rate conversion.

Figure 1:  A DSP audio system with a TMS320C542 signal processor. The CS4226 audio CODEC is connected using the BSP. Data is transferred to and from the host computer via HPI.

• The frequency-domain approach based on discrete Fourier transformation (DFT) or on frequency selective filtering with interpolation by M and then decimation by N
  
  
• The time-domain approach based on direct interpolation in time, in other words, on the realization of a sequence of non-integer delays.

You can also mix these two approaches, resulting in a reduction of the required intermediate sampling rate. In this article, we concentrate on the frequency-domain approach you implement with filtering.

Figure 1 shows a block diagram of a simple system to realize sampling-rate conversion. Output samples are calculated using difference equations, which utilize the up- and down-sampling and the filtering in-between. Table 1 presents the respective up- and down-sampling factors for the sampling rate conversions under consideration. These factors are equal to the least common multiple of a pair of sampling rates, divided by the respective rate of this pair.

Figure 2:  Sampling-rate conversion using an up-sampler with factor M, a low-pass filter, and a down-sampler with factor N

Table 1:  Sampling rate conversion factors

Since the up- and down-sampling factors for the rate conversion to/from 44.1 ksamples/s are too large, we accepted a slightly lower sampling rate, 44 ksamples/s (Table 2). This introduces a small error with a relative value of d = 0.22676% which, in most cases, is inaudible.

Table 2:  Sampling rate conversion to and from 44 ksamples/s

You can realize the filtering operation using an interpolator and decimation by a low-pass filter with gain M and a normalized cutoff frequency w_c=min(p/M, p/N) . You can design the respective FIR filter using the Parks-McClellan algorithm (Figure 3). Table 3 presents the lengths of FIR filters and the numbers of effective taps depending on the converted rates and the desired signal resolution (16- or 20-bit word length, corresponding to the stop-band attenuation of 96 or 120 dB, respectively).

Figure 3:  Illustrative frequency responses of FIR filters for two conversions: 32 to 48 ksamples/s and 32 to 44 ksamples/s

Table 3:  Low-pass FIR filters

Interpolation in Time Domain
You can calculate the output signal of arbitrary sampling rate using the time-domain approach. The following relationships determine the output signal sample:

where T_in = t_i+1 - t_i input sampling interval, T_out - output sampling interval, t_i - instant for a new output sample. These relationships are illustrated in Figure 4.

Figure 4:  Time relationship between input (red) and output (blue) samples

One of the simplest time-domain sampling-rate conversion methods consists of high oversampling and then choosing an appropriate output sample with the nearest time position to the required position. Figure 5 presents a multi-stage approach for this high oversampling. The time-analysis unit controls interpolators with factors 64 and 128, which computes the appropriate sampling instants. The main advantage of this approach (in comparison with the considered frequency-domain method) is the possibility of using the same filter coefficients for different sampling-rate ratios and, thus, a simplified realization of the interpolation filters.

Figure 5:  Multistage oversampling and choosing the output sample nearest to the required position in time

Time domain conversion can also be based on typical numerical methods such as polynomial interpolation, Lagrange interpolation, or spline interpolation. In the case of an Nth-order spline with a function defined in interval [x_k,..., x_k+m] as

6th-order interpolation is used with simple FIR filter to compensate sync⁷-distortion in the frequency domain caused by the spline interpolator.

Figure 6:  The sampling-rate conversion procedure consists of producing M data blocks from the received N data blocks, where M and N are the up- and down-sampling factors, respectively

The filter output is calculated only for these taps, for which input samples are non-zero. Using FIRS instruction and indirect addressing, the total number of multiplications is equal to L/(2M), where L is the filter length (Table 2). Table 4 presents the time requirements for conversion to and from the sampling rate of 48 ksamples/s, assuming 25 ns instruction cycles and 16-bit precision. For 60 samples, the conversion procedure needs 3180 cycles. From Table 4 we conclude that for 16-bit precision, a single DSP can additionally compute an FFT, which needs about 33,000 cycles for the 512-point FFT.

Table 4:  Time requirements during conversion to and from 48 ksamples/s