ADSP - LEC 2

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SP503Advanced Digital Signal Processing

Course Instructor: Engr. Touseef A. Rajput

Week # 2

25-10-2010

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December 30, 2010 SP503-Advanced Digital SignalProcessing

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Lecture 01 Review An analog signal is continuous in both time and amplitude. Analog signals in the

real world include current, voltage, temperature, pressure, light intensity, and so on.

The digital signal is the digital values converted from the analog signal at thespecified time instants.

Analog-to-digital signal conversion requires an ADC unit (hardware) and a low-passfilter attached ahead of the ADC unit to block the high-frequency components that

ADC cannot handle.

The digital signal can be manipulated using arithmetic. The manipulations mayinclude digital filtering, calculation of signal frequency content, and so on.

The digital signal can be converted back to an analog signal by sending the digital

values to DAC to produce the corresponding voltage levels and applying a smoothfilter (reconstruction filter) to the DAC voltage steps.

Digital signal processing finds many applications in the areas of digital speech andaudio, digital and cellular telephones, automobile controls, communications,biomedical imaging, image/video processing, and multimedia.

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Signal Sampling and Quantization

Below is an analog signal (solid line) defined at every point over the time

axis and amplitude axis. Hence, the analog signal contains an infinitenumber of points.

Why Sampling?

It is impossible to digitize an infinite number of points.

Furthermore, the infinite points are not appropriate to be processed by the

digital signal (DS) processor or computer, since they require infinite amount

of memory and infinite amount of processing power for computations.

Sampling can solve such a problem by taking samples at the fixed time

interval.

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Each sample maintains its voltage level during the sampling interval T to

give the ADC enough time to convert it.

This process is called sample and hold.

Since there exists one amplitude level for each sampling interval, we can

sketch each sample amplitude level at its corresponding sampling time

instant.

For a given sampling interval T, which is defined as the time span between

two sample points, the sampling rate is therefore given byfs = 1 / T samples per second (Hz).

For example, if a sampling period is T = 125 microseconds, the sampling

rate is determined as fs = 1 / (125 Qsec) = 8,000 samples per second (Hz).

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Next, we have to ensure that samples are collected at a rate high enoughthat the original analog signal can be reconstructed or recovered later.

If an analog signal is not appropriately sampled, aliasingwill occur, whichcauses unwanted signals in the desired frequency band.

The sampling theorem guarantees that an analog signal can be in theoryperfectly recovered as long as the sampling rate is at least twice as largeas the highest-frequency component of the analog signal to be sampled.The condition is described as fs >= fmax, where fmax is the maximum-

frequency component of the analog signal to be sampled.

Forexample

The minimum sampling rate to sample a speech signal (which containsfrequencies up to 4 kHz) is at least 8 kHz, or 8,000 samples per second.

To sample an audio signal, possessing frequencies up to 20 kHz, at least40,000 samples per second, or 40 kHz, of the audio signal are required.

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We call the 10-Hz sine wave the aliasing noise in this case, since the sampled amplitudes actually come from sampling the 90-Hz sine wave

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December 30, 2010 SP503-Advanced Digital SignalProcessing 7

Now let us develop the sampling theorem in frequency domain:

Consider a sampled signal xs(t) obtained by sampling the continuous signal

x(t) at a sampling rate of fs samples per second.

Mathematically, this process can be written as the product of the continuoussignal and the sampling pulses (pulse train):

xs(t) = x(t)p(t) (1)

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From spectral analysis, the original spectrumX( f ) and the sampled signal

spectrumXs( f ) in terms of Hz are related as

(2)

whereX( f ) is assumed to be the original baseband spectrum, whileXs( f ) is its

sampled signal spectrum, consisting of the original baseband spectrumX( f )

and its replicasX( f nfs).

Expanding Equation (2) leads to the sampled signal spectrum in Equation (3):

(3)

Equation (3) indicates that the sampled signal spectrum is the sum of thescaled original spectrum and copies of its shifted versions, called replicas.

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From the above figure, it is clear that the sampled signal spectrum consists of the scaled

baseband spectrum centered at the origin and its replicas centered at the frequencies of nfs

(multiples of the sampling rate) for each of n =1,2,3, . . . .

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If applying a low-pass reconstruction filter to obtain exact reconstruction

of the original signal spectrum, the following condition must be satisfied:

(4)

Solving Equation (4) gives

(5)

In terms of frequency in radians per second, Equation (5) is equivalent to

(6)

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which is a Fourier series expansion for a continuous periodic signal in terms

of the exponential form. We can identify the Fourier series coefficients as

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Signal Reconstruction Two simplified steps are involved:

First, the digitally processed data y(n) are converted to the ideal impulse trainys(t), in which each impulse has its amplitude proportional to digital output y(n),

and two consecutive impulses are separated by a sampling period of Tsecond, the analog reconstruction filter is applied to the ideally recoveredsampled signal ys(t) to obtain the recovered analog signal.

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b. Based on the spectrum in (a), the samplingtheorem condition is satisfied; hence, we can recover

the original spectrum using a reconstruction low-pass

filter. The recovered spectrum is shown as:

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a. The spectrum for the sampled signal is sketched

below:b. Since the maximum frequency of the analog

signal is larger than that of the Nyquist frequency,

the sampling theorem condition is violated.The recovered spectrum is shown below, where

we see that aliasing noise occurs at 3 kHz.

ADSP - LEC 2

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