- 2.1 The Time Domain
- 2.2 Sine Waves in the Frequency Domain
- 2.3 Shorter Time to a Solution in the Frequency Domain
- 2.4 Sine Wave Features
- 2.5 The Fourier Transform
- 2.6 The Spectrum of a Repetitive Signal
- 2.7 The Spectrum of an Ideal Square Wave
- 2.8 From the Frequency Domain to the Time Domain
- 2.9 Effect of Bandwidth on Rise Time
- 2.10 Bandwidth and Rise Time
- 2.11 What Does Significant Mean?
- 2.12 Bandwidth of Real Signals
- 2.13 Bandwidth and Clock Frequency
- 2.14 Bandwidth of a Measurement
- 2.15 Bandwidth of a Model
- 2.16 Bandwidth of an Interconnect
- 2.17 The Bottom Line

## 2.5 The Fourier Transform

The starting place for using the frequency domain is being able to convert a waveform from the time domain into a waveform in the frequency domain. We do this with the Fourier Transform. There are three types of Fourier Transforms:

- Fourier Integral (FI)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)

The Fourier Integral (FI) is a mathematical technique of transforming an ideal mathematical expression in the time domain into a description in the frequency domain. For example, if the entire waveform in the time domain were just a short pulse, and nothing else, the Fourier Integral would be used to transform to the frequency domain.

This is done with an integral over all time from - infinity to + infinity. The result is a frequency-domain function that is also continuous from 0 to + infinity frequencies. There is a value for the amplitude at every continuous frequency value in this range.

For real-world waveforms, the time-domain waveform is actually composed of a series of discrete points, measured over a finite time, T. For example, a clock waveform may be a signal from 0 v to 1 v and have a period of 1 nsec and a repeat frequency of 1 GHz. To represent one cycle of the clock, there might be as many as 1000 discrete data points, taken at 1-psec intervals. An example of a 1-GHz clock wave in the time domain is shown in Figure 2-4.

Figure 2-4 One cycle of a 1-GHz clock signal in the time domain (top) and frequency domain (bottom).

To transform this waveform into the frequency domain, the Discrete Fourier Transform (DFT) would be used. The basic assumption is that the original time-domain waveform is periodic and repeats every T seconds. Rather than integrals, just summations are used so any arbitrary set of data can be converted to the frequency domain using simple numerical techniques.

Finally, there is the Fast Fourier Transform (FFT). It is exactly the same as a DFT, except that the actual algorithm used to calculate the amplitude values at each frequency point uses a trick of very fast matrix algebra. This trick works only if the number of time-domain data points is a power of two, for example 256 points, or 512 points, or 1024 points. The result is a DFT, only calculated 100–10,000 times faster than the general DFT algorithm, depending on the number of voltage points.

In general, it is common in the industry to use all three terms, FI, DFT, and FFT, synonymously. We now know there is a difference between them, but they have the same purpose—to translate a time-domain waveform into its frequency-domain spectrum.

An example of a simple time-domain waveform and its associated spectrum, calculated by using a DFT, is shown in Figure 2-4.

At least once in his or her life, every serious engineer should calculate a Fourier Integral by hand, just to see the details. After this, we never again need to do the calculation manually. We can always get to an answer faster by using one of the many commercially available software tools that calculate Fourier Transforms for us.

There are a number of relatively easy-to-use, commercially available software tools that calculate the DFT or FFT of any waveform entered. Every version of SPICE has a function called the .FOUR command that will generate the amplitude of the first nine frequency components for any waveform. Most versions of the more advanced SPICE tools will also compute the complete set of amplitude and frequency values using a DFT. Microsoft Excel has an FFT function, usually found in the "engineering add-ins."