Husheng Li, UTK-EECS, Fall 2012 DISCRETE-TIME SIGNAL PROCESSING LECTURE 4 (SAMPLING) PERIODIC SAMPLING ο Sampling: π₯ π = π₯π (ππ), where T is the sampling period. In practice, it is done by A/D converter. The sampling operation is generally invertible. TWO STAGE REPRESENTATION We represent the sampling procedure in two stages: • Multiplication with an impulse train π = ∞ −∞ πΏ(π‘ − ππ) with output π₯π π‘ = π₯π π‘ π π‘ . • Conversion from impulse train to discrete time sequence Note: this is a mathematical formulation, not a physical circuit implementation FREQUENCY-DOMAIN REPRESENTATION ο The frequency domain of the post-sampling signal is given by ππ ππ€ ∞ 1 = ππ (π(π€ − ππ€π )) π π=−∞ ο ο Assume that the signal has a limited band −π€π , π€π . If the sampling frequency satisfies π€π ≥ 2π€π , there will be no overlap. EXACT RECOVERY ο An ideal low pass filter can be used to obtain the exact original signal. ALIASING If the inequality π€π ≥ 2π€π is not valid, the frequency copies of signal will overlap, which incurs a distortion called aliasing. ο See the example of cosine function. ο NYQUIST-SHANNON THEOREM ο Theorem: For a band limited signal within band −π€π , π€π , it is uniquely determined by its samples π₯π (ππ), if π€π ≥ 2π€π . EXAMPLE OF SINUSOIDAL SIGNAL RECONSTRUCTION OF A BANDLIMITED SIGNAL ο The reconstruction is given by π₯π π‘ ∞ = π=−∞ sin(π π‘ − ππ /π) π₯(π) π π‘ − ππ /π INTUITIVE EXPLANATION It can be used for D/C converter: ππ ππ€ = π»π ππ€ π(ππ€) ο DISCRETE-TIME PROCESSING ο We can use C/D converter to convert a continuous-time signal to a discrete-time one, process it in a discrete-time system, and then convert it back to continuous time domain. EXAMPLE: LTI AND LPF ο We can use a discrete-time low pass filter (LPF) to do the low pass filtering for continuous time signal. EXAMPLE: LTI AND LPF ο The ideal low pass discrete-time filter with discrete-time cutoff frequency w has the effect of an ideal low pass filter with cutoff frequency w/T. CONTINUOUS-TIME PROCESSING OF DISCRETETIME SIGNALS ο We can also use continuous-time system to process discrete-time signals. RESAMPLING: DOWNSAMPLING ο The downsampling π₯π π = π₯(ππ) implies 1 ππ ππ€ = π π−1 π=0 π€ π(π( − 2ππ/π)) π INTUITION IN THE FREQUENCY DOMAIN With aliasing Without aliasing DECIMATOR ο A general system for downsampling by a factor of M is the one shown above, which is called a decimator. UPSAMPLING ο The upsampling is given by π₯π π = π₯(π/πΏ), where L is the integer factor. EXPANDER The output of expander is given by π₯π π = ∞ −∞ π₯(π)πΏ(π − ππΏ . ο In the frequency, we have ππ π€ = π π€πΏ . ο INTERPOLATOR ο It can be shown that the above structure realizes the upsampling and interpolates the signals between samples: ∞ sin(π(π − ππΏ)/πΏ) π₯π π = π₯(π) π(π − ππΏ)/πΏ −∞ SIMPLE AND PRACTICAL INTERPOLATION ο The ideal interpolator is impossible to implement. In practice, we can use a linear interpolator: π β π = 1 − πΏ , |π| ≤ πΏ 0, ππ‘βπππ€ππ π TIME AND FREQUENCY OF LINEAR INTERPOLATOR CHANGING SAMPLING RATE BY A NON-INTEGER FACTOR ο The change of sampling rate by a non-integer factor can be realized by the cascade of interpolator and decimator. THE FREQUENCY INTUITION MULTIRATE SIGNAL PROCESSING ο Multirate techniques refer in general to utilizing upsampling, downsampling, compressors and expanders in a variety of ways to improve the efficiency of signal processing systems. INTERCHANGE OF FILTERING WITH COMPRESSOR / EXPANDER ο The operations of linear filtering and downsampling / upsampling can be exchanged if we modify the linear filter. MULTISTAGE DECIMATION ο ο The two stage implementation is often much more efficient than a single-stage implementation. The same multistage principles can also be applied to interpolation DIGITAL PROCESSING OF ANALOG SIGNALS ο In practice, continuous time signals are not precisely band limited, ideal filters cannot be realized, ideal C/D and D/C converters can only be approximated by A/D and D/A converters. PREFILTERING TO AVOID ALIASING ο We can use oversampled A/D to simplify the continuous-time antialiasing filter. FREQUENCY DOMAIN INTUITION ο Key point: the noise is aliased; but the signal is not. Then, the noise can be removed using a sharp-cutoff decimation filter. A/D CONVERSION SAMPLE-AND-HOLD The zero-order-hold system has the impulse response given by 1, 0 < π‘ < π β0 π‘ = 0, ππ‘βπππ€ππ π ο QUANTIZATION This quantizer is suitable for bipolar signals. ο Generally, the number of quantization levels should be a power of tow, but the number is usually much larger than 8. ο ILLUSTATION D/A CONVERSION ο The ideal D/A is given by ∞ sin(π π‘ − ππ /π) π₯π π‘ = π₯(π) π π‘ − ππ /π π=−∞ In practice, we need to use the above structure. OVERSAMPLING ο ο Oversampling can make it possible to implement sharp cutoff antialiasing filtering by incorporating digital filtering and decimation. Oversampling and subsequent discrete-time filtering and downsampling also permit an increase in the step size of the quantizer, or equivalently, a reduction in the number of bits required in the A/D conversion.