- 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.
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.
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.
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.
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.