Signal Integrity Engineer's Companion: The Wireless Signal--Part V

Fast Fourier Transform Analysis
The Fast Fourier Transform is at the heart of real-time spectrum analysis. In the RTSA, FFT algorithms are employed to transform time-domain signals into their frequency-domain spectral components. Conceptually, FFT processing can be considered as passing an RF signal through a bank of parallel filters where each filter has an equal frequency resolution and bandwidth but detects a different component. The FFT output generally is complex-valued, which means that it represents each frequency component as a vector with a particular phase and magnitude. For spectrum analysis, the amplitude of the complex component is usually of most interest.

The FFT process starts with properly decimated and filtered baseband I and Q components, which form the complex representation of the signal, with I as its real part and Q as its imaginary part. In FFT processing, a set of samples of the complex I and Q signals are processed at the same time. This set of samples is called the FFT frame. The number of samples in the FFT, generally a power of 2, is also called the FFT size. For example, a 1,024-point FFT can transform 1,024 I samples and 1,024 Q samples into 1,024 complex frequency-domain points. FFT Properties
A lot of jargon is associated with an FFT, which can obscure the simple concepts of an FFT process. Put simply, the amount of time required to acquire a set of samples upon which an FFT is performed is called the frame length. The frame length in the RSA is the product of the FFT size and the sample period; it's the time taken to perform a single measurement. Any changes, or temporal events, that occur in the measured signal during the frame length cannot be resolved.

Therefore, the frame length is simply the time resolution of the FFT process or a single measurement period for an RTSA. The frequency domain points of FFT processing are often called FFT bins. Therefore, the FFT size is equal to the number of bins in one FFT frame. Those bins are equivalent to the individual filter output of an RTSA parallel filter. All bins are spaced equally in frequency. Two spectral lines closer than the bin width cannot be resolved. The FFT frequency resolution therefore is the width of each frequency bin, which is equal to the sample frequency divided by the FFT size. Therefore, for the same sample rate, a larger FFT size yields a finer frequency resolution. For example, an RTSA with a sample rate of 25.6 MHz and an FFT size of 1,024 gives a frequency resolution of 25 kHz.

Frequency resolution can be improved by increasing the FFT size or by reducing the sampling frequency. The RTSA typically uses a digital down converter (DDC) and decimator to reduce the effective sampling rate as the frequency span is narrowed. This effectively trades time resolution for frequency resolution while keeping the FFT size and computational complexity at manageable levels. This approach allows fine resolution on narrow spans without excessive computation time on wide spans, where coarser frequency resolution is sufficient. The practical limit on FFT size is often display resolution, because an FFT with resolution much higher than the number of display points does not provide any additional information on the instrument's screen.

Windowing
An assumption inherent in the mathematics of Discrete Fourier Transforms (DFTs) and FFT analysis says that the data to be processed is a single period of a periodically repeating signal. Figure 10-8 depicts the FFT of a series of time domain frames. For example, when FFT processing is applied to Frame 2, the transform process assumes that the signal is periodic. However, the process generally produces discontinuities between successive frames, as shown in Figure 10-9. These artificial discontinuities generate spurious responses not present in the original signal, which can make it impossible to detect small signals in the presence of nearby large ones. This effect is called spectral leakage.