Mathematical Analysis of Sampling and Aliasing of Bandwidth-Limited Signals

introduction

Modern applications often require sampling analog signals, converting them into digital signals, then processing them, and finally reconstructing them into analog signals. The main problem discussed in this paper is how to sample and reconstruct analog signals while maintaining all the information of the original signal.

Bandwidth-limited signal

Let's start with finite-bandwidth signals. Doing so depends on both mathematical and physical considerations, which are explained below. A signal is called finite-bandwidth if its spectrum amplitude is zero except for a certain frequency point (the cutoff frequency). g(f) in Figure 1 is such a signal, with zero spectrum amplitude for frequencies above the frequency point a. In this case, a is also the bandwidth (BW) of the baseband signal. (Since negative frequencies have no physical meaning, the bandwidth of a baseband signal is defined only for positive frequencies.)

Figure 1. Spectrum of signal g(f)

Next, we sample g(f). We can express this operation mathematically as g(f) multiplied by a sequence of impulse functions separated by a time interval of T. By multiplying g(f) by the impulse function, we obtain the value of g(f) corresponding to the moment when the impulse function occurs, and the product is zero at any other time. This is similar to sampling g(f) at a frequency of f _SAMPLING = 1/T. This operation can be expressed as Equation 1, and the new signal after sampling is called s(t):

The next step is to find the spectrum of the sampled signal s(t). By taking a Fourier transform of equation 1, we get:

The above integral is a bit complicated to calculate. To simplify the calculation, note that s(t) is the product of g(f) and the impulse pulse train. We also know that multiplication in the time domain corresponds to convolution in the frequency domain. (For a proof of this conclusion, refer to any material on Fourier transform.) Therefore, S(f) can be expressed as:

Note that the asterisk in formula 3 indicates convolution, not multiplication. We already know the spectrum g(f) of the original signal, so we only need to calculate the Fourier transform of the impulse function sequence. We know that the impulse function sequence is a periodic function, so it can be represented by a Fourier series. As shown below:

The Fourier coefficients are:

The upper and lower limits of the integral in formula 5 are specified as only one period. This is no problem when dealing with impulse functions. However, in order to make the above expression more universal, the following substitution can be performed: replace the integral with a Fourier integral from negative infinity to positive infinity, and replace the periodic impulse function sequence with a single impulse function-the basic signal of the periodic signal. Therefore, formula 5 can be rewritten as:

In this way, the impulse function sequence can be expressed in the following simplified form, which is easy to Fourier transform:

Considering that a signal can be obtained from its Fourier transform integration, as follows:

and:

The final expression is as follows:

Based on the above results, reconsider the sampled baseband signal. Its Fourier transform expression is as follows:

The convolution of two signals A(f) and B(f) is defined as:

Then S(f) can be expressed as:

The result of the calculation is Equation 13, which is often called the sampling theorem. It shows that a signal sampled in the time domain with a period of T (seconds) will repeat the spectrum of the original signal at a frequency of 1/T, as shown in Figure 2. This result in turn can clearly and intuitively answer the previous question: How to sample an analog signal so that all the information of the original signal can be preserved?

Figure 2. Spectrum of the sampled signal s(t)

Aliasing Effect

To preserve all the information of the original baseband signal, it is necessary to ensure that each repeated spectrum "profile" does not overlap. If it overlaps (a phenomenon called aliasing), it will be impossible to recover the original signal from the sampled signal. This will cause high-frequency components to be aliased into the low-frequency band, as shown in Figure 3 .

Figure 3. Effects of aliasing on a signal.

To avoid aliasing, the following conditions must be met: 1/T > 2 , or 1/T > 2BW. This conclusion can also be expressed in terms of sampling frequency:

Therefore, the minimum sampling frequency that does not cause aliasing is 2BW. This is known as the Nyquist theorem.

Figure 3 shows the sampled signal that produces aliasing. Note that the high-frequency signal component f _H appears as a low-frequency component. You can use a low-pass filter to restore the original spectrum and filter out (attenuate) the other spectral components. When a low-pass filter with a cutoff frequency of is used to restore the signal, it cannot filter out the aliased high-frequency signal, resulting in degradation of the useful signal.

Considering that aliasing will degrade the useful signal, let's consider a specific limited bandwidth signal such as a bandpass signal. The low-frequency boundary of the bandpass signal is not zero. As shown in Figure 4 , the signal energy of the bandpass signal is distributed between _L and > _U , and its bandwidth is defined as _U - _L. Therefore, the main difference between a bandpass signal and a baseband signal lies in their bandwidth definition: the bandwidth of a baseband signal is equal to its highest frequency, while the bandwidth of a bandpass signal is the difference between the highest frequency and the lowest frequency.

Figure 4. Bandpass signal

From the previous discussion, we know that the sampling signal repeats the spectrum of the original signal with a period of 1/T. Because this spectrum actually includes a zero-amplitude band from 0Hz to the low-frequency cutoff frequency of the original bandpass signal, the actual signal bandwidth is lower than _U. Therefore, a certain frequency offset can be made in the frequency domain, allowing the sampling frequency to be lower than the sampling frequency required when the signal spectrum occupies the entire zero to _{U range. For example, assuming that the signal bandwidth is U} /2, the sampling frequency is taken as _U to satisfy the Nyquist theorem, and the spectrum of the sampled signal is shown in Figure 5 .

Figure 5. Spectrum of a bandpass sampled signal.

The sampling process does not produce aliasing, so if there is an ideal bandpass filter, the original signal can be completely recovered from the sampled signal. In this case, it is very important to note the difference between baseband and bandpass signals. For baseband signals, the bandwidth and corresponding sampling frequency are determined only by the highest frequency. The bandwidth of a bandpass signal is usually smaller than the highest frequency.

The above characteristics determine the method of recovering the original signal from the sampled signal. For baseband signals and bandpass signals with the same highest frequency, as long as a suitable bandpass filter is used to isolate the original signal spectrum (the white rectangular part in Figure 5), the bandpass signal can use a lower sampling frequency. Since the signal spectrum includes the shaded part, the low-pass filter used for baseband signal recovery cannot recover the original bandpass signal in this case, as shown in Figure 5. Therefore, if a low-pass filter is used to recover the bandpass signal in Figure 5, the sampling frequency must be above 2 _U to avoid aliasing.

A bandwidth-limited signal can only be fully recovered if the Nyquist criterion is satisfied. For a bandpass signal, the Nyquist sampling frequency can avoid aliasing only when a bandpass filter is used. Otherwise, a higher sampling frequency must be used. This is very important when choosing the converter sampling frequency in practical applications.

Another thing to note is the assumption of finite bandwidth signals. Mathematically, a signal cannot be truly finite bandwidth. The Fourier transform law tells us that if the duration of a signal is finite, its spectrum will extend to an infinite frequency range, and if its bandwidth is finite, its duration is infinite. Obviously, we cannot find a time domain signal that lasts infinitely, so there is no true finite bandwidth signal. However, the spectrum energy of most actual signals is concentrated within a finite bandwidth, so the previous analysis is still valid for these signals.

Sampling a sine signal

Sampling a sinusoidal signal is a very simple and convenient way to demonstrate the inherent phenomenon of high-frequency components appearing as low-frequency components when aliasing occurs. The spectrum of a pure sinusoidal signal only includes spikes (impulse functions) at the corresponding frequencies, and when aliasing occurs, the spikes will move from one frequency point to another.

The following results were obtained using the MAX19541 125Msps, 12-bit ADC. Figure 6 shows the frequency spectrum of the converter output signal when the input signal frequency f _IN = 11.5284MHz. The data shows that the main peak occurs exactly at this frequency. There are some other peaks in the spectrum, which are harmonics caused by the nonlinearity of the converter and are not related to the topic of this article. Since the sampling frequency f _SAMPLING = 125MHz is much larger than the twice input frequency required by the Nyquist theorem, no aliasing occurs.

Figure 6. Spectrum of a signal sampled by the MAX19541 ADC. f _SAMPLING = 125MHz, f _IN = 11.5284MHz.

Next, consider what happens to the position of the main peak if the input frequency is increased to f _{IN = 183.4856 MHz. This input frequency is greater than f SAMPLING} /2, so aliasing can be expected. Figure 7 shows the resulting spectrum, with the main peak falling at 58.48 MHz, which is the aliased signal. In other words, a frequency signal appears at 58.48 MHz that is not present in the original signal. Note that Figures 6 and 7 only show the spectrum below the Nyquist frequency, because the spectrum is periodic and the displayed portion of the figure already contains all the necessary information.

Figure 7. Spectrum of a signal sampled by the MAX19541 ADC. f _SAMPLING = 125MHz, f _IN = 183.4856MHz.