Decoding DTMF: Filters in the Frequency Domain

In Lab 5, you examined the behavior of several different filters. Some of the filters were “smoothing filters” that averaged the signal over many samples. Others were “sharpening” filters that accentuated transitions and edges. While it is very useful to understand the effects of these filters in the time-domain or (for images) the spatial-domain, it is often not easy to quantify these effects, especially when we are dealing with more complicated filters. Thus, just as we did with signals, we would like to understand the behavior of our filters in the frequency-domain. Assuming that our filter is linear and time-invariant, we can talk about the filter having a frequency response. We derive the frequency response in the following way. We know that if we put a complex exponential signal into such a filter, the output will be a scaled and shifted complex exponential signal with the same frequency. The amount of scaling and time shifting, though, is dependent on the frequency of the input signal. If we send a complex exponential signals with some frequency through the filter, we can measure the scaling and time shifting of that signal. The collection of complex numbers which corresponds to this scaling and shifting for all possible frequencies is known as the filter’s frequency response. The magnitude of the frequency response at a given frequency is the filter’s gain at that frequency. In this lab, we will be using the frequency response of filters to examine the problem solved by telephone touch-tone dialing. The problem is this: given a noisy audio channel (like a telephone connection), how can we reliably transmit and detect phone numbers? The solution, which was developed at AT&T, involves the transmission of a sum of sinusoids with particular frequencies. In order for this solution to be feasible, we must be able to easily decode the resulting signal to determine which numbers were dialed. We will see that we can do this easily by considering filters in the frequency domain.