MP2: Sampling and Interpolation

In this lab, you'll create sinusoids at a variety of sampling rates, both above and below Nyquist, and compare the resulting stem plots. Then you'll downsample a sinusoid, and interpolate it back to the original sampling rate, using a variety of different interpolation kernels.

In order to make sure everything works, you might want to go to the command line, and run

pip install -r requirements.txt

This will install the modules that are used on the autograder, including numpy, h5py, and the gradescope utilities.

Part 1: Comparing sinusoids at different frequencies

In this first part of the MP, you will create a sinusoid, and sample it at different sampling rates. The four parameters will be

The returned signal should be $$x[n] = \Re\{ z e^{j2\pi nf/F_s}\}$$

First, let's generate a sinusoid with plenty of samples, so we make sure it plots correctly. We'll use stem to plot it, so we can see each sample.

Now let's see what happens when we gradually lower the sampling rate.

Notice that, as long as you have at least two samples per period ($F_s\ge 2f$), you can reconstruct the original cosine (e.g., there are four sign-reversals in one second, so you can see that it's a 4Hz sinusoid).

When $F_s < 2f$, you can no longer see the real underlying cosine. It has been "aliased" to a new "alias frequency," $f_a=F_s-f$. That's because, for any integer $n$,

$$\cos\left(\frac{2\pi f n}{F_s}\right) = \cos\left(\frac{2\pi (F_s-f) n}{F_s}\right)$$

The highest frequency at which the cosine can be is called the "Nyquist rate", and it is $$F_N = \frac{1}{2}F_s$$

Let's adjust the sampling frequency so that 4Hz is just barely above or just barely below Nyquist.

Notice that, when $f > F_s/2$, it looks as though the sinusoid has a frequency of $F_s-f$. This is called its "aliased frequency." When the sampling rate gets even lower, aliasing can be even worse. The aliased frequency in general is

$$f_a = \min\left(f \mbox{mod} F_s, (F_s-f)\mbox{mod} F_s\right)$$

where $\mbox{mod}$ means modulo (written in python as f % Fs). Let's see some examples.

Exactly the same thing happens to sine waves as to cosines, except that:

  1. When $f=F_s/2$ exactly, the sine wave disappears.
  2. When $F_s/2 < f < F_s$, the sign of the sine reverses. That's because $$\sin\left(\frac{2\pi f n}{F_s}\right) = -\sin\left(\frac{2\pi (F_s-f) n}{F_s}\right)$$

In order to plot sine waves, we'll use a phasor of $-j=e^{-j\pi/2}$, to give it a phase of $-\pi/2$.

Part 2: Computing the alias frequency and aliased phasor

These two equations: $$\cos\left(\frac{2\pi f n}{F_s}\right) = \cos\left(\frac{2\pi (F_s-f) n}{F_s}\right)$$ $$\sin\left(\frac{2\pi f n}{F_s}\right) = -\sin\left(\frac{2\pi (F_s-f) n}{F_s}\right)$$

...can be combined to give this equation: $$\Re\left\{z \exp\left(j\frac{2\pi f n}{F_s}\right)\right\}=\Re\left\{z^* \exp\left(j\frac{2\pi (F_s-f) n}{F_s}\right)\right\}$$

In other words,

For example, let's look at the signal $$x(t) = \cos\left(2\pi ft-\frac{\pi}{4}\right)$$

This is a cosine delayed by $\pi/4$ radians. It's positive for about $3/8$ of a period, then negative for $1/2$ period, and so on.

Sometimes it's useful to explicitly calculate the aliased frequency and aliased phasor (HINT: you probably want to use the np.mod method):

If we plot a sinusoid at the aliased frequency and aliased phasor, it should look exactly the same as the original sinusoid.

Part 3: Fourier analysis

Remember that, if you have a signal that's periodic with a period $T_0$, then you can find its Fourier series coefficients using the Fourier analysis formula: $$X_k = \frac{1}{T_0}\int_{0}^{T_0} x(t) e^{-j\frac{2\pi kt}{T_0}}dt$$

We can get a pretty good measurement of $X_k$ on a computer by setting the sampling rate high enough, replacing the integral by a sum, and multiplying the integrand by $F_s$ (to represent $dt=1/F_s$). After simplifying a little, we get: $$X_k = \frac{1}{N_0} \sum_{n=0}^{N_0-1} x[n]e^{-j\frac{2\pi kn}{N_0}}$$ where $$N_0 = T_0F_s$$ During the rest of this MP, it will be useful to have a good Fourier analysis function, so let's write it.

Remember that, if a signal is periodic with period $N_0$, then it's also periodic with any period that's a multiple of $N_0$. We can use that fact to make the plot easier to read, e.g., by using four periods of a cosine as if they were just one period; that way the coefficient $X_1$ will be plotted as if it were $X_4$.

Let's choose a cosine with a really long period, say, 50 samples, and then compute the Fourier series coefficients over a space of 200 samples.

Part 4: Triangular kernel

Let's define the triangle function as follows: $$h[n]=\begin{cases}1-\frac{|t|}{T}&-T+1\le n\le T-1\\0&\mbox{otherwise}\end{cases}$$

This function will be useful when we try to interpolate signals, in part 5. Go ahead and write a function to compute it. Notice that you only need to include the nonzero samples, not the zero-valued samples.

Part 5: Interpolation

We are now going to simulate a discrete-to-continuous time converter by downsampling our 50-sample cosine by a factor of 25, and then upsampling it by the same factor of 25. The upsampling operation will use the same four types of interpolation that are used in typical D/C converters: rectangle, triangle, spline, and sinc.

First, let's downsample:

Now let's pretend we're interpolating it into continuous time. The general formula for interpolation is

$$x(t) = \sum_{n=-\infty}^\infty x[n] h\left(t-nT\right)$$

where $T$ is the sampling period.

In order to simulate D-to-C conversion, we will use $t$ to represent time in the highrate signal (e.g., $0\le t<200$) instead of time in the continuous-time signal. We'll use $n$ to represent time in the lowrate signal (e.g., in the example, $0\le n< 8$), while $T$ is the ratio between the two sampling rates (e.g., $T=25$).

Notice those last few lines from the docstring. The triangle function has a length of $2T-1$. The lowrate signal has a length of $N$. If you put in a triangle function in place of every lowrate sample, you'd get a total length of $NT+(T-1)$, but that's not what we want -- what we want is a signal of length $NT$. In order to make that happen, we'll use modulo indexing, i.e., the interpolation formula should be implemented as

for n in range(N):

$$x\left((t+nT) \mbox{mod} NT\right) += h(t) x[n]$$

You might find the function np.mod to be useful.

Well, it doesn't look very much like a cosine! In fact, it's just a linear interpolation between the lowrate samples.

In order to better understand what's going on, let's look at its spectrum. In the following plot you should see that most of the spectral energy is at $X_1$, but there is significant nonzero energy also at $X_3$, $X_5$, and so on:

Part 6: Rectangle interpolation

Rectangle interpolation is much, much worse than triangle interpolation, but it's much easier to implement in hardware, so it's actually much more common in the real world. Let's try it, and see how it turns out: