Rf and if digitization in Radio Receivers: Theory, Concepts, and Examples
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| 3.3 Algorithms
Providing a general discussion on algorithms used for implementing radio receiver functions is difficult. This is due to the wide variety of types of receivers as well as the various ways of implementing the required receiver operations for each type of receiver. In short, algorithms are highly application-specific. The details of the many potential algorithms used in radio receivers are beyond the scope of this report. It is beneficial, however, to look at an example algorithm to observe the methodology in determining algorithm complexity vs. the potential for real-time operation. This type of assessment is crucial for radio receivers that use digitization at the RF or IF. The FFT is an example of an algorithm frequently used in digital signal-processing and radio receiver applications. The FFT transforms time-domain samples of received signals into a set of frequency-domain samples, allowing operations on the received signals to be performed directly in the frequency domain. These received signals are typically bandpass signals when digitization occurs at the RF or IF. These bandpass signals may be digitized using either sampling at twice the maximum frequency, bandpass sampling, or oversampling. The FFT also can be applied to signals that have been downconverted to baseband but the signal must be split into co-phase and quadrature-phase components before digitization unless coherent downconversion has been used. Regardless of the sampling method employed, the resolution of the transformed signal in the frequency domain is a function of both the time spacing between the samples of the signal Δt in Data Block 0 Data Block 2 Data Block 3 Processing Time for Data Block 0
t Data Block 1 Processing Time for Data Block 1
Processing Time for Data Block 2 Processing Time for Data Block 3
Processor 1 Processor 2 … …
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the time domain and the number of samples N used in the computation of the FFT. The frequency spacing between samples in the frequency domain is then given as
Δf = 1 N Δt (Hz) . The maximum frequency of the spectrum is then
max =
N 2 Δf = 1 2Δt since there are N/2 samples in the FFT computed from N real-valued time-domain samples. (Actually, there are (N/2)+1 samples in the FFT ranging from DC to f max if both DC and f max are
included.) For a fixed N-point FFT (i.e., an FFT computed from N real-valued time-domain samples), the frequency spacing between the frequency domain samples Δf must be changed in order to change the maximum frequency of the spectrum. That requires changing the time spacing Δt between the time-domain samples. By decreasing Δt and holding N constant, the time duration of a block of N samples is reduced and the maximum frequency of the spectrum is thereby increased. Therefore, for fixed values of N, the higher the maximum frequency desired, the shorter the duration of the block of N samples must be. With a fixed number of samples N, a given processor, and a given FFT algorithm, 3 the processing time to compute an N-point FFT is fixed. For real-time operation, this computation of the FFT (including any other required processing such as windowing and any required data transfers) must be performed within the time taken to capture all N samples of the current data block assuming that a single processor is used. A parameter called real-time bandwidth then can be defined as the maximum frequency that can be processed in real time. To achieve real-time processing, careful consideration of processor speed, the signal bandwidth (data rate), the number of computations required in implementing the signal-processing algorithms, and the speed of any necessary data transfers is required. An example showing how to estimate the amount of processing power required for real-time analysis is given below. For this example, the simplified case of looking at the time required to compute an FFT is investigated. While this example shows the methodology used to determine a required processing speed, the processing required for radio receiver applications normally would involve much more than computing a single FFT. In this simplified example, it is assumed that an input signal is sampled at a fixed rate and that the only processing performed on the sampled input signal is the FFT. No other processing (such as windowing or averaging) is performed. Data transfer time is also neglected. Assuming a bandlimited input signal with a maximum frequency of 5 MHz, the 2f
sampling rate would be 10 Msamples/s. For this sampling rate, the time between samples Δt is 100 ns. Assume that FFTs are computed from blocks of N = 1024 samples. Therefore, it takes NΔt = 102.4 μs to capture a block of data. The number of floating-point operations (actually multiplications) required 3 to compute an N-point FFT is estimated as N log 2 N. Therefore,
3 There are many different FFT algorithms available requiring different numbers of floating-point operations. N log 2 N commonly is used as an approximation for the number of floating-point operations required in computing an FFT. 31
roughly 10,240 floating-point operations are required for the 1024-point FFT. In order to achieve real-time processing in the single processor case, the FFT must be computed within the time period required to capture a block of data (102.4 μs). The minimum required processing speed is then found by m f f a a s q m a a f a a = m m ss s P In this simplified case, the minimum processing speed is 100 MFLOPS. One can then compare the required processing speed to the processing speeds available for different types of processors such as those listed in Table 3. |