# Husheng Li UTKEECS Fall 2012 DISCRETETIME SIGNAL PROCESSING

- Slides: 36

Husheng Li, UTK-EECS, Fall 2012 DISCRETE-TIME SIGNAL PROCESSING LECTURE 4 (SAMPLING)

PERIODIC SAMPLING �

TWO STAGE REPRESENTATION

FREQUENCY-DOMAIN REPRESENTATION �

EXACT RECOVERY � An ideal low pass filter can be used to obtain the exact original signal.

ALIASING �

NYQUIST-SHANNON THEOREM �

EXAMPLE OF SINUSOIDAL SIGNAL

RECONSTRUCTION OF A BANDLIMITED SIGNAL �

INTUITIVE EXPLANATION �

DISCRETE-TIME PROCESSING � 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 � We can use a discrete-time low pass filter (LPF) to do the low pass filtering for continuous time signal.

EXAMPLE: LTI AND LPF � 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 � We can also use continuous-time system to process discrete-time signals.

RESAMPLING: DOWNSAMPLING �

INTUITION IN THE FREQUENCY DOMAIN With aliasing Without aliasing

DECIMATOR � A general system for downsampling by a factor of M is the one shown above, which is called a decimator.

UPSAMPLING �

EXPANDER �

INTERPOLATOR �

SIMPLE AND PRACTICAL INTERPOLATION �

TIME AND FREQUENCY OF LINEAR INTERPOLATOR

CHANGING SAMPLING RATE BY A NON-INTEGER FACTOR � 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 � 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 � The operations of linear filtering and downsampling / upsampling can be exchanged if we modify the linear filter.

MULTISTAGE DECIMATION 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 � 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 � We can use oversampled A/D to simplify the continuous-time antialiasing filter.

FREQUENCY DOMAIN INTUITION � 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 �

QUANTIZATION � This quantizer is suitable for bipolar signals. � Generally, the number of quantization levels should be a power of tow, but the number is usually much larger than 8.

ILLUSTATION

D/A CONVERSION �

OVERSAMPLING � � 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.

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