Sampling and mixing analysis of limited bandwidth signals

1 Introduction

More and more applications require sampling analog signals, converting them into digital signals, performing various calculations and processing on digital signals, and then converting them into analog signals. This article discusses how to sample analog signals and shaping them to preserve the original signal.

2 baseband signal sampling and mixing analysis

Starting with the finite bandwidth signal, the finite bandwidth signal refers to the signal with a Spectrum Composition of 0 for all frequencies beyond a certain frequency point (cutoff frequency point. G (t) in 1, all the spectrum components in the frequency range greater than the cut-off frequency alpha are 0. In this case, α is the bandwidth of the baseband signal. The mathematical expression of sampling g (t) can be expressed by multiplying g (t) by the impulse function with a period of T. G (t) is sampled for the signal value at the excitation point, and the value of other points is 0. From the analog signal point of view, it is to sample g (t) by frequency fsampling = 1/T. The sampled signal S (t) can be expressed using the following formula:

To obtain the spectrum of the sampled signal S (t), Fourier transformation can be performed on S (t:

The impulse string function is a periodic function, which can be expressed by Fourier series, as shown in the following formula:

The Fourier coefficient here is:

In the above formula, the upper and lower limits of points are determined by only one cycle. Under the premise of ensuring the equivalence, the following transformations can be performed: Replace the points in the above formula with the points from negative infinity to positive infinity, and the periodic Impulse Function is replaced by the fundamental frequency impulse function, the above formula can be rewritten:

The impulse string function can be expressed in the simplified form of Fourier transformation in the following ways:

Considering that a signal can be obtained by its Fourier transform integral, the following formula is used:

The final result is as follows:

Based on the above results, reconsider the fundamental frequency signal, and its Fourier transformation is:

Convolution of two signals a (F) and B (f) is defined:

Then S (f) can be rewritten:

The above formula is what we often call the sampling law. It indicates that the signal sampled by period t will repeat the spectrum of the original signal at a 1/T frequency in the time domain, as shown in figure 2.

To retain the information of all original signals, it is necessary to ensure that there is no overlap between each repeated spectrum. Otherwise, it is impossible to restore the original signal from the sampling signal. Mixing means that the high frequency section masks the low frequency section signal, as shown in 3. To avoid mixing, the following conditions must be met: 1/T ≥2α or 1/T ≥2bw. The sampling frequency can also be expressed as follows: We should also note the assumption of limited bandwidth signals. From a mathematical analysis, a signal cannot be truly limited in bandwidth. Fourier's law tells us that if a signal is limited in the time domain, its spectrum will be extended to infinity, and if its bandwidth is limited, then it is infinite in the time domain. Obviously, we cannot find a time-domain signal with an infinite period, so it is impossible to have a real limited bandwidth signal. However, the spectrum energy of most actual signals is concentrated in limited bandwidth, so the previous analysis is still effective for these signals. The sampled sine signal can be very simple and convenient to detect whether the sampling frequency is low, because the mixing phenomenon is characteristic of the low sampling frequency. The peak in the Sine Signal Spectrum (impulse string function) only appears at the corresponding frequency point. When there is an alias, the peak will move to another frequency point, which corresponds to the mixing signal.

Fsampling ≥ 2bw

The above results indicate that the minimum sampling frequency for non-aliasing is 2bw. This is the nequest sampling law.

Figure 3 shows the sampled signal that is overlapped. The high-frequency signal component FH is superimposed on the low-frequency part. During design, a low-pass filter is usually used to restore the original spectrum and filter out other spectrum components. When a low-pass filter with a cut-off frequency of α is used to restore the signal in figure 3, it cannot filter out the high-frequency signals that are overlapped, resulting in signal deterioration.

3. Analysis of sampling and mixing of band-pass Signals

Let's look at another limited bandwidth signal, band-pass signal. The low frequency cutoff point of the band-pass signal is not 0Hz. In Figure 4, the spectrum energy of a band-pass signal ranges between α L and α U, and its bandwidth is defined as α U-α L. The main difference between band-pass signals and baseband signals is the definition of bandwidth. The bandwidth of the baseband signal is equal to its high-frequency cut-off frequency, while the bandwidth of the band-pass signal is equal to the difference between the high-frequency cut-off frequency and the low-frequency cut-off frequency. We can see from the previous discussion that the sampled signal repeats the spectrum of the original signal in a 1/T period. Because the spectrum actually includes a zero-amplitude band from 0Hz to the low-frequency cut-off frequency of the original band-pass signal, the actual bandwidth of the original band-pass signal is smaller than that of α U. In this way, a certain frequency offset can be performed in the frequency field, and the sampling frequency can be reduced. To satisfy the nequest law, an original band-pass signal with an actual bandwidth of α U/2 can be sampled at a frequency of α U, as shown in spectrum 5 of the sampled signal. Such sampling does not produce aliasing. Therefore, if an ideal band-pass filter is available, the original signal can be completely restored. In this example, the differences between baseband and band-pass signals are very important. For baseband signals, the bandwidth and the corresponding sampling frequency are only determined by the high-frequency cutoff point. The bandwidth of band-pass signals is usually lower than the high-frequency cutoff frequency.

4. sampling method and result analysis

The above features determine the different methods for restoring the original signal from the sampling signal. For baseband and band-pass signals with the same high-frequency cutoff point, as long as a suitable band-pass filter is adopted, the sampling frequency of band-pass signals can be reduced (the White Rectangle in Figure 5 ). In this case, the low-pass filter cannot recover the original signal. As shown in figure 5, the shadow is still included in the recovery signal spectrum. Therefore, if a low-pass filter is used to restore the band-pass signal in figure 5, the sampling frequency must be above 2α u to avoid mixing. A limited-bandwidth signal can be completely restored only when the nequest law is met. For Band-pass signals, when using a band-pass filter, the use of the NAI quest sampling frequency can avoid mixing. Otherwise, a higher sampling frequency is required. This is important when ADC and DAC are selected in practical applications.

The following test results are obtained using the 125 MSPs and 12-bit ADC: max19541 recently released by Maxim. Figure 6 shows its output signal spectrum. The input signal frequency is fin = 11.5284 MHz. Obviously, the highest peak occurs at this frequency point. There are other small spikes in the spectrum attention, which are caused by the harmonic wave caused by the nonlinear nature of the ADC, and are irrelevant to the topic discussed in this article. Because the sampling frequency fsample = 125 MHz is much more than twice the input signal frequency required by the nequest law, there is no mixing phenomenon. If the input frequency is increased to fin = 183.4856 MHz, greater than fsample/2, there should be a mix of them. Figure 7 shows the output spectrum of fin> fsample/2. The main peak falls at 58.48mhz, which is an alias signal. That is to say, there is a signal not included in the original signal at 58.48mhz. In Figure 6 and Figure 7, only the spectrum below the nequest frequency is given. Because the spectrum is periodic, the display section in the figure already contains all necessary information.

Figure 6 and Figure 7

5 conclusion

The above test results show that the sampling law is the basic tool for signal sampling, and strict mathematical analysis is also very important for parameter selection.