Chapter 13 Frequency Analysis

In the frequency domain, you can conceptually separate the sine waves that add to form the complex time-domain signal. Figure 13-1 shows single frequency components, which spread out in the time domain, as distinct impulses in the frequency domain. The amplitude of each frequency line is the amplitude of that frequency component’s time waveform.

Some measurements, such as harmonic distortion, are very difficult to quantify by inspecting the time waveform on an oscilloscope. When the same signal is displayed in the frequency domain by an FFT Analyzer, also known as a Dynamic Signal Analyzer, you easily can measure the harmonic frequencies and amplitudes.

Aliasing

According to Shannon’s sampling theorem, the highest frequency (Nyquist frequency: fN) that can be analyzed is fN = fs/2, where fs is the sampling frequency. Any analog frequency greater than fN after sampling appears as a frequency between 0 and fN. Such a frequency is known as an alias frequency. In the digital (sampled) domain, there is no way to distinguish these alias frequencies from the frequencies that actually lie between 0 and fN. Therefore these alias frequencies need to be removed from the analog signal before sampling by the A/D converter.

In order to remove these components present at frequencies higher than the Nyquist frequency, you must use an analog lowpass filter. The anti-aliasing analog lowpass filter should exhibit a flat passband frequency response with a good high-frequency alias rejection and a fast roll-off in the transition band.

FFT Fundamentals

The Fast Fourier Transform (FFT) is a fast version of the Discrete Fourier Transform (DFT). The DFT transforms digital time domain signals into the digital frequency domain.

Figure 13-2. FFT Transforms Time-Domain Signals into the Frequency Domain

Chapter 13 Frequency Analysis

Each frequency component is the result of a dot product of the time domain signal with the complex exponential at that frequency and is given by the equation:

n = 0

The DC component is the dot product of x(n) with [cos(0) – jsin(0)], or with 1.0.

The first bin, or frequency component, is the dot product of x(n) with cos(2π n/N) – jsin(2π n/N). Here, cos(2π n/N) is a single cycle of the cosine wave, and sin(2π n/N) is a single cycle of a sine wave.

In general, bin k is the dot product of x(n) with k cycles of the cosine wave for the real part of X(k) and the sine wave for the imaginary part of X(k).

The use of the FFT for frequency analysis implies two important relationships.

The first relationship links the highest frequency that can be analyzed to the sampling frequency and is given by the equation:

fs

Fmax = --

2

where Fmax is the highest frequency that can be analyzed, and fs is the sampling frequency.

Refer to the Aliasing section in this chapter for more information about

Fmax.

The second relationship links the frequency resolution to the total acquisition time, which is related to the sampling frequency and the block size of the FFT and is given by the equation: