# Sampling

## Ideal Sampling

In the Fourier Domain,

### Nyquist Theorem

then our reconstructed signal will be

This is why we call reconstructing a signal from its samples "sinc interpolation." This leads us to formulate the Nyquist Theorem.

### Discrete Time Processing of a Continuous Time Signal

Assuming that the Nyquist criteria is met holds,

This shows us that as long as the Nyquist theorem holds, we can process continuous signals with a disrete time LTI system and still have the result be LTI.

### Continuous Time Processing of Discrete Time Signals

While not useful in practice, it can be useful to model a discrete time transfer function in terms of Continuous Time processing (e.g a half sample delay).

Similar to the analysis of DT processing of a CT signal, we can write the discrete transfer function in terms of the continuous function. Our continuous signal will be bandlimited after reconstruction.

### Downsampling

### Upsampling

When we upsample a signal by a factor of L, we are interpolating between samples. Conceptually, this means we are reconstructing the original continuous time signal and resampling it at a faster rate than before. First we place zeros in between samples, effectively expanding our signal.

## Multi-Rate Signal Processing

Notice that we only need one LPF to take care of both anti-aliasing and interpolation.

### Exchanging Filter Order During Resampling

### Polyphase Decomposition

Now each of our filters is compressible, so we can switch the order of downsampling and filtering while maintaining the same output.

Now for any filter, we can compute only what we need, so the result is correct and efficently obtained.

## Practical Sampling (ADC)

Unfortunately, ideal analog to digital conversion is not possible for a variety of reasons. The first is that not all signals are bandlimited (or there may be noise outside of the bandwidth). Moreover, computers only have finite precision, so we cannot represent the full range of values that a continuous signal might take on with a finite number of bits per sample. The solution to the first issue is to include a “anti-aliasing” filter before the sampler. The solution to the second issue is to quantize.

However, sharp analog filters are difficult to implement in practice. To deal with this, we could make the anti-aliasing filter wider, but this would add noise and interference. If we keep the cutoff frequency the same, then we could alter part of the signal because our filter is not ideal. A better solution is to do the processing in Discrete Time because we have more control. We also sample higher than the Nyquist Rate and then downsample it to the required rate.

### Quantization

We do this under the following assumptions:

This means our Signal to Noise Ratio for quantization is

## Practical Reconstruction (DAC)

In the ideal case, we reconstruct signals by converting them to impulses and then convolving with a sinc. However, impulses are require lots of power to generate, and sincs are infinitely long, so it is impractical to design an analog system to do this. Instead, we use an interpolation like Zero-Order-Hold to pulses and then filter with a reconstruction filter.

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