# Basics of Band-Limited Sampling and Aliasing

Abstract: This article presents a theoretical approach for sampling and reconstructing a signal without losing the original contents of the signal. The effects of aliasing are covered. The MAX19541 ADC is used as an example for comparing over- and under-sampled input frequencies.

*RF Design*magazine.

## Introduction

## Band-Limited Signals

**Figure 1**, we see that the spectrum is zero for frequencies above α. In that case, the value α is also the bandwidth (BW) for this baseband signal. (The bandwidth of α baseband signal is defined only for positive frequencies, because negative frequencies have no meaning in the physical world.)

*Figure 1. Frequency spectrum of the signal G(f).*

_{SAMPLING}= 1/T. This operation is expressed in Equation 1, and the new sampled signal is called s(t):

**Figure 2**shows our result in graphic form. That result, in turn, now allows us to answer the original question clearly and intuitively: How do we sample in a way that preserves the full information of the original signal?

*Figure 2. Frequency spectrum of the sampled signal, s(t).*

## The Effects of Aliasing

**Figure 3**.

*Figure 3. Diagram of a signal affected by aliasing.*

_{H}, appears at a much lower frequency. You can recover a signal from its sampled version by using a lowpass filter to isolate the original spectrum, and cutting (attenuating) everything else. Thus, extracting the signal with a lowpass filter of cutoff frequency α does not eliminate the aliased high frequency, but allows it to corrupt the signal of interest.

**Figure 4**has signal energy between the frequencies α

_{L}and α

_{U}, and its bandwidth is defined as α

_{U}- α

_{L}. Thus, the main difference between bandpass and baseband signals is in their definition of bandwidth: The bandwidth of a baseband signal equals its highest frequency, while the bandwidth of a bandpass signal is the difference between its upper- and lower-bound frequencies.

*Figure 4. Illustration of a bandpass signal.*

_{U}. We can, therefore, use smaller shifts in the frequency domain, which allows a sampling frequency lower than that required for a signal whose spectrum occupies all frequencies from zero to α

_{U}. Assume, for example, a signal bandwidth of α

_{U}/2. To satisfy the Nyquist criterion our sampling frequency equals α

_{U}, producing the sampled-signal spectrum of

**Figure 5**.

*Figure 5. Spectrum of sampled bandpass signal.*

_{U}to avoid aliasing.

## Sampling Sinusoidal Signals

**Figure 6**shows the spectrum at the converter's output for the input frequency f

_{IN}= 11.5284MHz. The data shows that the main spike occurs exactly at this frequency. There are a number of other spikes which are harmonics introduced by the nonlinearities of the converter, but they are irrelevant to our discussion. The sampling frequency, f

_{SAMPLING}= 125MHz, is more than twice the input frequency as required by the Nyquist criterion, and therefore no aliasing occurs.

*Figure 6. Spectrum of signal sampled with MAX19541 ADC. Here f*

_{SAMPLING}= 125MHz, f_{IN}= 11.5284MHz._{IN}= 183.4856MHz. This input frequency is higher than f

_{SAMPLING}/2, and we expect aliasing to occur. The resulting spectrum given in

**Figure 7**shows that the main spike is now located at 58.48MHz, and this is the aliased signal. In other words, an image has appeared at 58.48MHz when, in fact, our input signal did not contain this frequency. Note that in both Figures 6 and 7 we plotted the spectrum only up to the Nyquist frequency. This is because the spectrum is periodic and this portion contains all the essential information.

*Figure 7. Spectrum of a signal sampled with MAX19541 ADC. Here f*

_{SAMPLING}= 125MHz, f_{IN}= 183.4856MHz.