FFT Analyzer. Gianfranco Miele, Ph.D

Transcription

1 FFT Analyzer Gianfranco Miele, Ph.D

2 Introduction It is a measurement instrument that evaluates the spectrum of a time domain signal applying the Fourier transform. + X f = x t e j2πft dt Observing this mathematical relation, it is possible to affirm that for an exact calculation of the frequency spectrum of an input signal, an infinite period of observation would be required. The result of this calculation would be a continuous spectrum, so the frequency resolution would be unlimited.

3 Introduction It is obvious that we cannot have an infinite period of observation but only a limited portion recorded in the time domain. To obtain the spectrum of the signal, the DFT, or better the FFT, can be applied to the recorded signal. N 1 X n f = x k e j2πnk N k=0 As a consequence, the frequency resolution is limited but the spectrum can nevertheless be determined with sufficient accuracy.

4 Simplified block diagram Analog signal f S x(t) Signal conditioning ADC Windowing Memory FFT Display Digital signal

5 Signal conditioning Goal It manipulates the analog signal in order to meet the requirements of the next block for further processing. As a consequence, to comprehend the operations that this stage has to execute, we have to analyze the characteristics of the next block. ADC

6 Analog-to-Digital Converter ADC Sampler Quantizer

7 Analog-to-Digital Converter Sampler It converts a continuous input signal into a time-discrete signal and the information about the time characteristic is lost. "Signal Sampling" by 4mobile (talk) Licensed under Public domain via Wikimedia Commons -

8 Analog-to-Digital Converter Sampler The bandwidth of the input signal must be limited or else the higher signal frequencies will cause aliasing effects due to sampling. no aliasing f max = f S 2 f S 2 f S 3f S 2 aliasing f max > f S 2 f S 2 f S 3f S 2

9 Analog-to-Digital Converter Quantizer It maps the amplitude values of the samples to a countable set. Generally a n-bit ADC has 2 n possible values. "3-bit resolution analog comparison" by Hyacinth Own work. Licensed under Creative Commons Attribution-Share Alike 3.0 via Wikimedia Commons

10 Analog-to-Digital Converter Quantizer The difference between the quantized amplitude value of the digital signal and the actual amplitude value of the analog signal is called quantization error.

11 Analog-to-Digital Converter Quantizer 2V fs /2 /2 The range of input values that produces the same output is called quantum ( ). = 2V fs 2 n = V fs 2 n 1 Therefore the quantization error has a uniform distribution ranging in 2, 2.

12 Analog-to-Digital Converter Quantizer 2V fs This error can be considered as a quantization noise with a RMS value /2 /2 V rms = x 2 dx = 12

13 Analog-to-Digital Converter Underloading In an underload condition, the input signal does not fill the entire range of the ADC's available dynamic range. All of the available dynamic range of the analog-to-digital converter is not used effectively.

14 Analog-to-Digital Converter Large DC bias If the input signal has a large DC bias, all the dynamic range is occupied by the DC signal. As a consequence the alternating part is greatly influenced by the quantization error.

15 Analog-to-Digital Converter Overloading The ADC's dynamic range is too low for the amplitude of the input signal and this causes the clipping of the signal.

16 Signal conditioning Signal Conditioning Coupling AC coupling is used to avoid quantization error due to a large DC bias. It removes the DC component of the input signal by using a high pass filter.

17 Signal conditioning Signal Conditioning Coupling Amplifier or Attenuator The incoming signal is amplified or attenuated in order to better exploit the ADC's dynamic range.

18 Signal conditioning Signal Conditioning Coupling Amplifier or Attenuator Anti-alias filter To limit the input signal bandwidth a Low-Pass Filter with a cut-off frequency lower than f S /2. From AN Agilent Vector Signal Analysis Basics Copyright Agilent Technologies

19 Memory The output of the ADC represents a band-limited, digital version of the analog input signal in time-domain. This digital data stream is stored in sample memory. It can be seen as a circular FIFO (first in, first out) buffer that collects individual data samples into blocks of data called time records. N-1 0

20 Windowing To mitigate the effect of the spectral leakage several weighting functions called windows are used. They heavily weight the beginning and end of the sample record to zero and the middle of the sample is weighted towards unity. Several window function have been designed and they have different spectral characteristics.

21 Windowing Rectangular window The ADC's dynamic range is too low for the amplitude of the input signal and this causes the clipping of the signal.

22 Windowing Rectangular window Narrow main lobe Very large side lobes Slow roll-off It can introduce large amplitude errors up to 36%.

23 Windowing Hanning window

24 Windowing Hanning window Main lobe larger than rectangular window main lobe Side lobes large but lower than rectangular window side lobe Roll-off fast It can introduce amplitude errors up to 16%.

25 Windowing Flat-top window

26 Windowing Flat-top window Main lobe is very large and flat Side lobes are very low It is very accurate for amplitude measurement but has a very poor frequency resolution.

27 FFT and display The windowed time records are used by the Digital Signal Processor (DSP) in order to compute the FFT and the result is displayed on a display.

28 FFT and display 0 f max Span

29 Zoom 0 f max f max 2

30 Zoom In order to enlarge the desired spectrum portion it is not sufficient to plot the previously-computed bins that are in the considered frequency interval, because we will have a poor resolution. To improve the frequency resolution we have to change the sampling rate. In fact, choosing f = f S N f S < f S we obtain f = f S N < f

31 New block diagram Analog signal f S x(t) Signal conditioning ADC Decimating filter Windowing Memory FFT Display Digital signal

32 Decimating filter Decimating filter Digital LPF Decimator The digital decimating filters simultaneously decrease the sample rate and limit the bandwidth of the signal. The sampling rate into the digital filter is f S ; the sampling rate out of the filter is f S /n, where n is the decimation factor and an integer value. Similarly, the bandwidth at the input filter is BW, and the bandwidth at the output of the filter is BW/n.

33 Decimating filter Decimating filter Digital LPF Decimator Typically a binary decimation is performed, which means that the sampling rate is changed by integer powers of 2. Span = Span 2 m These new span are called cardinal spans.

34 Zoom 0 f max 2

35 Arbitrary span In order to obtain arbitrary span it is necessary to interpolate and resample the time record stored in the memory. In this way we can have any desired sampling rate.

36 New block diagram Analog signal f S x(t) Signal conditioning ADC Decimating filter Windowing Resampling Memory FFT Display Digital signal

37 Band selectable Fourier analysis The FFT is inherently a baseband transform. This means that the frequency range of the FFT starts from 0 Hz (or DC) and extends to some maximum frequency, f S /2. This can be a significant limitation in measurement situations where a small frequency band, that does not start from 0 Hz, needs to be analyzed.

38 Band selectable Fourier analysis 0 f C f max 2 f max

39 New block diagram Analog signal f S e j2πf CnT S x(t) Signal conditioning ADC Decimating filter Windowing Resampling Memory FFT Display Digital signal

40 Band selectable Fourier analysis The frequency span of interest is mixed with a complex sinusoid at the zoom span center frequency, which causes that frequency span to be mixed down to baseband. The signal is filtered and decimated/resampled for the specified span and all out-of-band frequencies are removed.

41 Band selectable Fourier analysis f C span 2 f C f C + span 2

42 Strong points The phase information is not lost during the complex Fourier transform. FFT analyzers are therefore able to determine the complex spectrum by magnitude and phase. It takes a time record of the signal and processes all frequencies simultaneously.

43 Weak points Quantization of the samples causes the quantization noise which causes a limitation of the dynamic range towards its lower end. The higher the resolution (number of bits) of the A/D converter used, the lower the quantization noise. Due to the limited bandwidth of the available highresolution A/D converters, a compromise between dynamic range and maximum input frequency has to be found for FFT analyzers. FFT analyzers are not suitable for the analysis of pulsed signals.