Can we replace the analog low-pass filter in Figure 2, which is often difficult to build to required accuracy, with a digital filter after the signal is quantized and coded? Strange as it may sound, because it looks as though we are defeating the very purpose of the filter, but it is done in some applications. However, this requires sampling the signal at much higher frequency than the Nyquist rate, to avoid aliasing in the analog signal, and reducing the sampling rate after digital filtering as shown in Figure 7.

This technique makes the life of designer easy, as sharp cut-off is no longer needed in the analog filter. Imagine designing an audio filter with the slope of the attenuation characteristic from 15 kHz to 16 kHz going down by about 60 dB to 80 dB. This is a typical requirement in a FM quality digital audio, which has a sampling rate of 32 kHz. When a signal is sampled at ƒ_s the spectra of the quantization noise occupies this band. When the signal is sampled at higher rate the same noise (which depends only on the number of bits used) spreads up to new ƒ_s as shown in Figure 8. As shown Figure 8(a), the noise density N up to ƒ_s when sampled at 2ƒ_s reduces to N/2 as shown in Figure 8(b) so the total noise power is same. In Figure 8(c) the sampling frequency is increased to 4ƒ_s and the noise density drops to N/4 keeping the total integrated noise power constant. We notice that the noise in the Nyquist band is reduced as shown by the rectangular window in Figure 8(a). It is this noise that contributes to the signal to noise ratio finally. The decrease of noise in the Nyquist band can be traded off against the number of bits used in quantization. We know from Equation 1 that reducing one bit increases the noise by 6 dB, which can be compensated by increasing the sampling rate by four times. This is the basic technique used in delta-sigma modulators that employ one bit quantizer at very high sampling rate compared to the Nyquist rate, followed by noise shaping and decimation filters to reduce the sample rate and increase the number of bits in the code word. The signal to quantization noise ratio considering the sampling frequency is given by

(S / N)_dB = 6.02n + 1.8 + 10log(ƒ_s / 2) (5)

Thus, each time the sampling rate is doubled the signal to noise ratio in the signal band increases by 3 dB.

In applications such as software-defined radio (SDR), the RF signals are directly digitized using undersampling technique and processed with DSPs. The signals occupying an RF band if sampled at Nyquist rate would need very high speed analog-to-digital converter (ADC) and very high speed DSP to process. This need not be the case. In fact, the sampling rate would be close to that given by Equation 2 in which ƒ_max is replaced by the bandwidth B of the signal. We use the concept of under sampling to sample bandpass signals. This concept can be easily understood by carrying out a simple experiment using a set up shown in Figure 9.

We use a variable frequency oscillator, a sampler at fixed sampling frequency and an oscilloscope. In fact, we can use a sampling oscilloscope replacing the sampler and the oscilloscope. We begin increasing the oscillator frequency starting from a low value and observe the sampled frequency in the oscilloscope. The observed frequency follows the oscillator frequency up to half the sampling frequency. For example, if the sampling scope has sampling rate of 50 MHz, then we observe the output on the scope monotonically increases up to 25 MHz. When the oscillator frequency is increased beyond 25 MHz we observe that the frequency on the oscilloscope starts decreasing, finally reaching zero (DC), when the frequency from the oscillator is exactly 50 MHz. When the input frequency is increased further the observed frequency on the oscilloscope starts increasing and goes up to 25 MHz as input frequency goes to 75 MHz. Beyond this it starts decreasing again. This phenomenon continues, the observed frequency always varying between DC and 25 MHz. We can characterize the behavior of the sampler with this experiment and plot a graph of the observed frequency versus the input frequency as shown in Figure 10.

As we see in this figure, the observed frequency is always between 0 and 0.5ƒ_s while input is varied from 0 to ƒ_s, ƒ_s to 2ƒ_s and so on. What we see is actually the alias frequency when the input frequency is more than 0.5ƒ_s. Thus, the frequencies ƒ₂, ƒ₃ and ƒ₄ are aliased to ƒ₁. From Figure 10 we realize that if a signal occupies a band of frequencies from nƒs to 0.5 (n+1) ƒ_s, as shown in Figure 11(a), after sampling ƒ_s they will be folded back to 0 to 0.5ƒ_s as shown in Figure 11(b) when n is odd and as shown in Figure 11(c) when n is even. (Reader may note that Figure 11 is a spectral diagram with x-axis showing the frequency and y-axis showing the amplitude, while Figure 10 is a plot of output vs. input frequency of sampler).

Thus, for a bandpass signal from lowest frequency f1 to the highest frequency f2, with bandwidth B = f2 - f1 the minimum sampling frequency is given by

min ƒ_s = 2ƒ₂ / N

where N is an integer part of the ratio f₂ /B.

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