# Chapter 11. Modulation and Sampling

In “Aliasing” we saw that when a signal is sampled at 10,000 Hz, a component at 5500 Hz is indistinguishable from a component at 4500 Hz. In this example, the folding frequency, 5000 Hz, is half of the sampling rate. But I didn’t explain why.

This chapter explores the effect of sampling and presents the Sampling Theorem, which explains aliasing and the folding frequency.

I’ll start by exploring the effect of convolution with impulses; I’ll use that effect to explain amplitude modulation (AM), which turns out to be useful for understanding the Sampling Theorem.

The code for this chapter is in `chap11.ipynb`, which is in the repository for this book (see “Using the Code”). You can also view it at http://tinyurl.com/thinkdsp-ch11.

# Convolution with Impulses

As we saw in “Systems and Convolution”, convolution of a signal with a series of impulses has the effect of adding up shifted, scaled copies of the signal.

As an example, I’ll read a signal that sounds like a beep:

```filename = '253887__themusicalnomad__positive-beeps.wav'
wave.normalize()```

And I’ll construct a wave with four impulses:

```imp_sig = thinkdsp.Impulses([0.005, 0.3, 0.6,  0.9],
amps=[1,     0.5, 0.25, 0.1])
impulses = imp_sig.make_wave(start=0, duration=1.0,
framerate=wave.framerate)```

and then convolve them:

`convolved = wave.convolve(impulses)`

Figure 11-1 shows the results, with the signal in the top left, the impulses in the lower left, and the result on the right. ...

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