# Transforms: Part 4 - Discrete Fourier Transforms

As we saw earlier when discussing transform duality, when something happens on one side of a transform, we can predict through duality what to expect on the other side.

Fig.1 - The fundamental frequency coefficient is calculated here. As the positive and negative parts of a sine wave only differ in polarity, it is more efficient to subtract pairs of samples and perform one multiplication.

Fig.2a) The FFT butterfly consists of summing and differencing a pair of samples. Fig.2b) The processing of Fig.1 is implemented here with butterflies.

Fig.3. The discrete spectrum of the FFT is redundant as it is mirrored about F4.

# Look Like A Million For Less

Every TV viewer compares live content with what they regularly see on TV, with multimillion-dollar talent with more multimillions in technical equipment and support.

# Information: Part 3 - Applying Statistics

We are not done with statistics yet. In a sense we will never be done with it and it is better to know how to deal with it than to ignore it. It is better still to know how others…

# Transforms: Part 5 - OFDM

Thus far we have looked at transforms from a somewhat abstract viewpoint. In contrast, here we look at an application where transforms take center stage.

# Electricity: Part 3 - AC Systems

Here we look at alternating current (AC) systems and how generating AC often requires an intermediate step of converting to DC to improve the efficiencies of AC generators.

# Information: Part 2 - Gaussian Distribution

Information can never be separated from its nemesis, which is uncertainty. The former is only possible by limiting the latter.