Contents
Introduction
With microprocessors becoming ever increasingly faster, smaller, and cheaper, it is preferable to use digital signal processing as a way to compensate for distortions caused by analog circuitry. One area that this can be applied is in signal reconstruction, where a low pass analog filter is used on the output of a digital-to-analog converter to attenuate unwanted frequency components above the Nyquist frequency.
The problem with analog low pass filters is that higher the order, the more resistors, capacitors, and op-amps are required in its construction. More circuit components means more circuit board space, which is a precious commodity with today's hand-held devices.
Here, it will be explained how up-sampling can be used to relax requirements on analog low pass filter design while decreasing signal distortion.
A Representative DT Signal
For this discussion, a representative signal $ x(t) $ will be used to demonstrate the process of signal reconstruction. We will look at the signal in the frequency domain, as shown in the plot below:
As seen in the plot, the signal $ X(f) $ has a triangular shape and is band-limited by $ f_{max} $. By the Nyquist-Shannon Sampling Theorem, the sampling frequency $ f_{s1} $ must be greater than $ 2f_{max} $. For this discussion, assume $ f_{s1} $ is just slightly greater than $ 2f_{max} $.
Now, we are going to sample $ x(t) $ with an impulse train with period $ T_{s1} = 1/f_{s1} $ and convert to a discrete time signal. This has the affect of scaling the magnitude axis of $ X(f) $ by $ 1/T_{s1} $ and the frequency axis by $ 2\pi T_{s1} $, and then repeating the result every $ \omega = 2\pi $. This yields the following plot $ X_d(\omega) $:
Without Up-sampling
First, consider the process of signal reconstruction without up-sampling. The following diagram shows the process:
The first block is a digital-to-analog converter. This converts the discrete time signal to a continuous time signal. This has the effect of scaling the magnitude axis by $ T_{s1} $ and the omega axis by $ \frac{1}{2\pi T_{s1}} $. This yields the following plot $ W_1(f) $:
The next step is to use an analog low pass filter to remove all frequency components above $ \frac{1}{2T_{s1}} $. The important note to take from $ W_1(f) $ is that the frequency gap for which the analog filter has to transition from low attenuation to high attenuation is $ \frac{1}{T_{s1}} - 2f_{max} $.
With Up-sampling
Now, consider the process of signal reconstruction with up-sampling. The following diagram shows the process:
The first block is the up-sample by $ N $ operator, where $ N\in \mathbb{N} $. This inserts $ N-1 $ zeros between each point in the time domain. In the DTFT domain, this has the effect of scaling the omega axis by $ 1/N $. This yields the following plot $ U_w(/omega) $.
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