CES 520 - WEEK 11 October 31, 2006 - Sampled Data

The Fourier series

• Any periodic signal may be decomposed into harmonic sinusoids of the proper amplitude and phase
• The Discrete Fourier Transform (DFT) generates the amplitudes of the harmonic sinusoids
• Multiplies each sine/cosine by the periodic signal and integrates over one period
• The real part is the cosine term, the imaginary part is the sine term
• The Fast Fourier Transform (FFT) is an algorithm to calculate the DFT quickly
• FFT calculation proportional to N log(N) versus N² for the DFT
• e.g. N=2¹⁶=65536: N/log(N) = 65536/16 = 4096x speed improvement

• The Fourier transform is also defined for non-periodic and non-sampled signals:
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• A waveform must be both sampled and periodic to have a finite numeric solution to the Fourier transform
• Fourier transform and inverse Fourier transform are the same (except for sign of exponent of e term)
• Time/frequency and Frequency/Time dualities:
• Repetition:
• Periodic <==> Sampled
• Non-periodic <==> Continuous
• Wide <==> Narrow
• Position offset <==> Change in phase
• Even symmetry <==> Real
• Odd symmetry <==> Imaginary
• Assymetrical <==> Complex

Digital signals

• Block diagram of a typical digital system
• May have analog input only, analog output only, or both
• Output sample rate may differ from input sample rate in a multirate system.
• Decimation: output sample rate = (1/N) * input sample rate
• Interpolation: output sample rate = N * input sample rate
• Resampling: input and output sample rates are not harmonically related
• A "digital" signal implies digitization in amplitude as well as time.
• Digitization in amplitude (quantization)
• Quantization effects
• Broadband quantization noise if the analog signal is "noise like", i.e. not discrete frequency(s)
• If the analog signal is sinusoidal, quantization "noise" will be discrete frequencies
• S/N ratio is roughly 6 dB per bit of resolution
• Practical ADCs are usually worse. Example: LTC2203 81.6 dB SNR, but 6*16 bit = 96 dB
• Digitization in time (sampling)
• Sampling in the time domain causes periodicity in the frequency domain
• The spectrum of the analog signal is repeated at the sample rate
• Nyquist frequency = sample frequency / 2
• Aliasing is the overlap of the frequency spectrum if the sample rate is too low
• Nyquist criterion:
• The analog signal must be band-limited to less than the Nyquist frequency to avoid aliasing.
• Sampling theorem
• If Nyquist criterion is met, the sampled signal can be perfectly reconstructed
• This ignores the effect of amplitude quantization (digitization).
• Sampled signal must be filtered (frequency domain) = interpolated (time domain)
• The ideal reconstruction filter is a rectangle function in the frequency domain.
• Cut-off frequency = Nyquist frequency = sample rate / 2
• The ideal reconstruction filter is a sinc function in the time domain:
• sinc(t) = sin(Pi*t)/(Pi*t)
• Can be viewed as filtering out the aliased frequencies or as interpolating between sample points
• Digital filters
• An infinitely-narrow impulse makes a good test signal because the spectrum is flat.
• The Fourier transform of the filter inpulse response is the filter's frequency spectrum.
• A Finite Impulse Response (FIR) filter's coefficients equal the impulse response.
• Linear system: filter output is the summation of the impulse response x input samples
• An FIR filter is a convolution engine.
• An Infinite Impulse Response filter has feedback
• More closely resembles an analog filter.
• Requires fewer multipliers but wider bit width.
• Not linear-phase. Group delay is not flat.
• Harder to design.
• Design of FIR filter coefficients
• Use your favorite filter design program
• Use the DFT method
• Specify desired response in the frequency domain.
• Generate time-domain coefficients with an inverse DFT
• Apply a window
• Do DFT on resulting impulse response to see if it meets frequency-domain requirements
• Change frequency response and iterate until requirements are met
• A so-called digital system needs two analog filters:
• Input low-pass filter to make sure input signal meets the Nyquist criterion
• Output low-pass filter to eliminate the repeated spectrum near the sampling frequency and harmonics

Analog interfacing

• D/A converters
• Pulse-width modulator
• Timer(s) control the period and duty factor (percent ON time).
• A low-pass filter integrates the pulses to give a smooth waveform.
• Good linearity but poor gain accuracy.
• Sigma delta modulator
• A "one-bit" DAC
• Pseudo-random sequence puts most energy at high frequencies where it can be filtered easily.
• Used in CD players.
• R-2R ladder
• Resolution vs sample rate
• Higher sample rate spreads quantization noise over broader bandwidth. Less noise per Hz.
• A/D converters
• Flash converter
• Fastest ADC type
• Low resolution, typically 6 or 8 bits
• Used for video
• Integrating
• Slow but excellent linearity
• Dual-slope type is more accurate - counts time to reset ramp using calibrated reference
• Used in DC voltmeters
• Sigma-delta converter
• Analagous to sigma-delta DAC
• The average value of the 1-bit output is proportional to the analog voltage.
• Successive approximation
• Most common type
• Good accuracy and speed

Assignments:

• Read Embedded Systems Design, Chapter 5, pp 141-156.