DISCRETE-TIME SIGNAL PROCESSINGLECTURE 4 (SAMPLING)

Husheng Li, UTK-EECS, Fall 2012

PERIODIC SAMPLING

Sampling: , 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 with output • Conversion from impulse

train to discrete time sequence

Note: this is a mathematical formulation, not a physical circuit implementation

FREQUENCY-DOMAIN REPRESENTATION

The frequency domain of the post-sampling signal is given by

Assume that the signal has a limited band .

If the sampling frequency satisfies , there will be no overlap.

EXACT RECOVERY

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

ALIASING

If the inequality is not valid, the frequency copies of signal will overlap, which incurs a distortion called aliasing.

See the example of cosine function.

NYQUIST-SHANNON THEOREM

Theorem: For a band limited signal within band , it is uniquely determined by its samples , if .

EXAMPLE OF SINUSOIDAL SIGNAL

RECONSTRUCTION OF A BANDLIMITED SIGNAL

The reconstruction is given by

INTUITIVE EXPLANATION

It can be used for D/C converter:

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 DISCRETE-TIME SIGNALS

We can also use continuous-time system to process discrete-time signals.

RESAMPLING: DOWNSAMPLING

The downsampling implies

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

The upsampling is given by , where L is the integer factor.

EXPANDER

The output of expander is given by .

In the frequency, we have

INTERPOLATOR

It can be shown that the above structure realizes the upsampling and interpolates the signals between samples:

SIMPLE AND PRACTICAL INTERPOLATION

The ideal interpolator is impossible to implement. In practice, we can use a linear interpolator:

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

The zero-order-hold system has the impulse response given by

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

The ideal D/A is given byIn practice, we need to use the above structure.

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.