Multirate DSP and Its Application in A/D Conversion

This article reviews the application of multirate DSP in achieving a more efficient A/D conversion and clarifies why we need different sampling rates within a single system.

In digital signal processing, we commonly need to change the sampling rate of the signal to achieve a more efficient system. Incorporating more than one sampling rate within a system is called multirate signal processing.
An ADC converts a continuous-time signal,

, into a digital sequence. To this end, it samples the input signal and quantizes the amplitude of each sample.

Periodic Sampling

The sampling operation can be mathematically modeled by first multiplying the continuous-time signal by an impulse train and then converting the result into a discrete-time sequence. The final result will be a discrete-time sequence

given by


,

where

is the sampling period and its reciprocal is the sampling frequency
. The sampling operation can be represented by a system referred to as an ideal continuous-to-discrete-time (C/D) converter. The block diagram of a C/D converter and the corresponding waveforms are shown in Figure 1.


Figure 1. A C/D converter multiplies the input by an impulse train s(t) and generates a discrete-time sequence. Image courtesy of Discrete-Time Signal Processing.

Note that, in Figure 1,

is still a continuous-time signal; however,
is a discrete-time sequence in which the x-axis is normalized to
.

The Fourier Transform of a Sampled Signal

As shown in Figure 1, during sampling operation, the input is multiplied by an impulse train and we have


Equation 1

Multiplication in the time domain corresponds to convolution in the frequency domain, and we obtain (Appendix, Equation A1)


 
Equation 2

where

and
denote, respectively, the frequency and the sampling frequency in radians/second.
and
represent the Fourier transform of
and
, respectively. Note that Equation 2 gives the Fourier transform of
, not that of
; however, for the purpose of this article, we don’t need to know the Fourier transform of
. Equation 2 shows an important relation between the Fourier transform of
and
. According to this equation, if we ignore the scaling factor
,
has replicas of
at multiples of
. This is illustrated in Figure 2.


Figure 2. Multiplying a signal by an impulse train leads to replicas of the input spectrum at multiples of the sampling frequency. Image courtesy of Discrete-Time Signal Processing.

The Nyquist Sampling Theorem

We want

to be a representation of
. The question is, can we reconstruct the original continuous-time signal from
? In other words, given the spectrum in Figure 2(c), can we obtain the frequency domain representation of
shown in Figure 2(a)?
Figure 2 suggests that we can reconstruct the original signal by applying a low-pass filter to

such that the frequency components below
are kept and replicas of
at
are removed. However, this is possible only if
, otherwise, there is no separation between the replicas and we cannot apply the required low-pass filtering. The condition
, which is often referred to as the Nyquist sampling theorem, prevents the replicas from overlapping with each other. The mentioned overlapping leads to a kind of distortion called aliasing distortion, or simply aliasing.
To successfully reconstruct

from
, we need
to be a band-limited signal; otherwise, aliasing will occur. For example, Figure 2(a) shows that
has all its energy at
, i.e.,
for
. In practice,
is not generally a band-limited signal. While we are mainly interested in a particular frequency band of
, there will be strong components or, at least, noise components at frequencies above the desired band. Hence, when sampling with
, we need to place a low-pass filter before the C/D to sufficiently attenuate all the frequency components above
. This filter which prevents aliasing is called an anti-aliasing filter.

Minimum Possible Sampling Rate Requires Very Sharp Filters

Suppose that we want to sample an analog music waveform where the desired energy band is in the range

(Figure 3(a)). According to the Nyquist sampling theorem, the minimum sampling frequency which can be used in this case is
; however, this requires an anti-aliasing filter with a very steep roll-off. The filter must pass the frequency components from zero to just below
and reject all the components above
(Figure 3(b)). The anti-aliasing filter, which is placed before the sampler, is an analog filter and, unfortunately, analog filters cannot achieve a flat passband along with a very sharp transition from passband to stopband. Therefore, in practice, we cannot use a sampling rate of
for this example.

Combined Analog and Digital Filter

The obvious solution for avoiding the use of a very sharp analog filter will be using a sampling rate higher than

. For example, suppose that we increase the sampling rate by a factor of
and use
. In this case, the stopband edge of the anti-aliasing filter will be
(Figure 3(c)). The passband is still the same as before and we need to pass the frequencies below
. As a result, the width of the filter’s transition band will be
, which is practical. Aliasing can be avoided in this way; however, the analog filter will not sufficiently suppress the frequency components from
to
, and these unwanted components will enter the system.


Figure 3. (a) The spectrum of the input signal. (b) The ideal anti-aliasing filter required when using
. (c) Increasing the sample rate relaxes the analog filter requirements. (d) The overall system which uses both analog and digital filtering. Image courtesy of IEEE.

Fortunately, after the ADC, we have the option of using a digital filter (Figure 3(d)), which can offer both sharp transition and linear-phase response. In this way, we can sufficiently suppress the unwanted components from

to
.
So far, our system is not a multirate one because there is only one sampling rate used in the system. The overall system obtained from two filters (the analog prefilter and the digital filter) and the analog-to-digital converter is equivalent to that obtained by a sharp analog anti-aliasing filter with passband edge of 22kHzand an ADC sampling at 88 kHz.
But is this system efficient? Do we really need to use

samples/second to represent a signal that does not have frequency components above
? Note that after the analog prefilter, there could still be frequency components between
and
, but these will be removed by the digital filter. And we know that, according to the Nyquist criterion, we only need
samples/second to represent our input signal, which has all its energy below
. This means that we can discard some of the output samples of the above system and still retain all the information we are interested in. Since we want to reduce the sampling rate from
to
, we can keep one sample from every two consecutive samples. This operation is called decimation or downsampling (by a factor of
).

Now there are two sampling rates in our system; before decimation, we were using a sampling rate of

, and after decimation, the sampling rate is
. Hence, we have a multirate system. This operation reduces the number of bits used to represent the input signal by a factor of
. See page 32 of CMOS Integrated Analog-to-Digital and Digital-to-Analog Converters to read about a simple trick which can be used to even further relax the requirements of the analog prefilter in Figure 3(d).

Decimation

A discrete-time sequence

that has been downsampled by a factor of
is given by the following expression:



This means that we are using only one sample out of every M consecutive samples. In other words, if the sampling rate of

was
, the sampling rate of
will be
. The symbol used for a factor-of-M decimator, and an example of factor-of-2 decimation is illustrated in Figure 4(a), and 4(b), respectively.


Figure 4. (a) The symbol used for factor-of-M decimation and (b) illustration of factor-of-2 decimation. Image courtesy of IEEE.

Since factor-of-M decimation is equivalent to sampling the underlying analog signal,

, with the sampling rate
, we obtain



According to the Nyquist criterion, if

has frequency components above
, aliasing will occur. As a result, we usually need to place a low-pass filter with stopband edge frequency of
before the factor-of-M decimation block. In the example of Figure 3, this filtering task is accomplished by the digital filter that precedes the factor-of-2 decimation stage. The normalized cutoff frequency of this filter will be
. This is illustrated in Figure 5.


Figure 5. (a) We need a band-limiting filter prior to decimation; (b) the filter used for the factor-of-M decimation. Image courtesy of IEEE.

Appendix


Equation A1



Previous
Next Post »
My photo

Hi, I`m Sostenes, Electrical Technician and PLC`S Programmer.
Everyday I`m exploring the world of Electrical to find better solution for Automation. I believe everyday can become a Electrician with the right learning materials.
My goal with BLOG is to help you learn Electrical.
Related Posts Plugin for WordPress, Blogger...

Label

KITAIFA NEWS KIMATAIFA MICHEZO BURUDANI SIASA TECHNICAL ARTICLES f HAPA KAZI TU. LEKULE TV EDITORIALS ARTICLES DC DIGITAL ROBOTICS SEMICONDUCTORS MAKALA GENERATOR GALLERY AC EXPERIMENTS MANUFACTURING-ENGINEERING MAGAZETI REFERENCE IOT FUNDAMENTAL OF ELECTRICITY ELECTRONICS ELECTRICAL ENGINEER MEASUREMENT VIDEO ZANZIBAR YETU TRANSDUCER & SENSOR MITINDO ARDUINO RENEWABLE ENERGY AUTOMOBILE SYNCHRONOUS GENERATOR ELECTRICAL DISTRIBUTION CABLES DIGITAL ELECTRONICS AUTOMOTIVE PROTECTION SOLAR TEARDOWN DIODE AND CIRCUITS BASIC ELECTRICAL ELECTRONICS MOTOR SWITCHES CIRCUIT BREAKERS MICROCONTROLLER CIRCUITS THEORY PANEL BUILDING ELECTRONICS DEVICES MIRACLES SWITCHGEAR ANALOG MOBILE DEVICES CAMERA TECHNOLOGY GENERATION WEARABLES BATTERIES COMMUNICATION FREE CIRCUITS INDUSTRIAL AUTOMATION SPECIAL MACHINES ELECTRICAL SAFETY ENERGY EFFIDIENCY-BUILDING DRONE NUCLEAR ENERGY CONTROL SYSTEM FILTER`S SMATRPHONE BIOGAS POWER TANZIA BELT CONVEYOR MATERIAL HANDLING RELAY ELECTRICAL INSTRUMENTS PLC`S TRANSFORMER AC CIRCUITS CIRCUIT SCHEMATIC SYMBOLS DDISCRETE SEMICONDUCTOR CIRCUITS WIND POWER C.B DEVICES DC CIRCUITS DIODES AND RECTIFIERS FUSE SPECIAL TRANSFORMER THERMAL POWER PLANT cartoon CELL CHEMISTRY EARTHING SYSTEM ELECTRIC LAMP ENERGY SOURCE FUNDAMENTAL OF ELECTRICITY 2 BIPOLAR JUNCTION TRANSISTOR 555 TIMER CIRCUITS AUTOCAD C PROGRAMMING HYDRO POWER LOGIC GATES OPERATIONAL AMPLIFIER`S SOLID-STATE DEVICE THEORRY DEFECE & MILITARY FLUORESCENT LAMP HOME AUTOMATION INDUSTRIAL ROBOTICS ANDROID COMPUTER ELECTRICAL DRIVES GROUNDING SYSTEM BLUETOOTH CALCULUS REFERENCE DC METERING CIRCUITS DC NETWORK ANALYSIS ELECTRICAL SAFETY TIPS ELECTRICIAN SCHOOL ELECTRON TUBES FUNDAMENTAL OF ELECTRICITY 1 INDUCTION MACHINES INSULATIONS ALGEBRA REFERENCE HMI[Human Interface Machines] INDUCTION MOTOR KARNAUGH MAPPING USEUL EQUIATIONS AND CONVERSION FACTOR ANALOG INTEGRATED CIRCUITS BASIC CONCEPTS AND TEST EQUIPMENTS DIGITAL COMMUNICATION DIGITAL-ANALOG CONVERSION ELECTRICAL SOFTWARE GAS TURBINE ILLUMINATION OHM`S LAW POWER ELECTRONICS THYRISTOR USB AUDIO BOOLEAN ALGEBRA DIGITAL INTEGRATED CIRCUITS FUNDAMENTAL OF ELECTRICITY 3 PHYSICS OF CONDUCTORS AND INSULATORS SPECIAL MOTOR STEAM POWER PLANTS TESTING TRANSMISION LINE C-BISCUIT CAPACITORS COMBINATION LOGIC FUNCTION COMPLEX NUMBERS ELECTRICAL LAWS HMI[HUMANI INTERFACE MACHINES INVERTER LADDER DIAGRAM MULTIVIBRATORS RC AND L/R TIME CONSTANTS SCADA SERIES AND PARALLEL CIRCUITS USING THE SPICE CIRCUIT SIMULATION PROGRAM AMPLIFIERS AND ACTIVE DEVICES BASIC CONCEPTS OF ELECTRICITY CONDUCTOR AND INSULATORS TABLES CONDUITS FITTING AND SUPPORTS CONTROL MOTION ELECTRICAL INSTRUMENTATION SIGNALS ELECTRICAL TOOLS INDUCTORS LiDAR MAGNETISM AND ELECTROMAGNETISM PLYPHASE AC CIRCUITS RECLOSER SAFE LIVING WITH GAS AND LPG SAFETY CLOTHING STEPPER MOTOR SYNCHRONOUS MOTOR AC METRING CIRCUITS APPS & SOFTWARE BASIC AC THEORY BECOME AN ELECTRICIAN BINARY ARITHMETIC BUSHING DIGITAL STORAGE MEMROY ELECTRICIAN JOBS HEAT ENGINES HOME THEATER INPECTIONS LIGHT SABER MOSFET NUMERATION SYSTEM POWER FACTORS REACTANCE AND IMPEDANCE INDUCTIVE RESONANCE SCIENTIFIC NOTATION AND METRIC PREFIXES SULFURIC ACID TROUBLESHOOTING TROUBLESHOOTING-THEORY & PRACTICE 12C BUS APPLE BATTERIES AND POWER SYSTEMS ELECTROMECHANICAL RELAYS ENERGY EFFICIENCY-LIGHT INDUSTRIAL SAFETY EQUIPMENTS MEGGER MXED-FREQUENCY AC SIGNALS PRINCIPLE OF DIGITAL COMPUTING QUESTIONS REACTANCE AND IMPEDANCE-CAPATIVE RECTIFIER AND CONVERTERS SEQUENTIAL CIRCUITS SERRIES-PARALLEL COMBINATION CIRCUITS SHIFT REGISTERS BUILDING SERVICES COMPRESSOR CRANES DC MOTOR DRIVES DIVIDER CIRCUIT AND KIRCHHOFF`S LAW ELECTRICAL DISTRIBUTION EQUIPMENTS 1 ELECTRICAL DISTRIBUTION EQUIPMENTS B ELECTRICAL TOOL KIT ELECTRICIAN JOB DESCRIPTION LAPTOP THERMOCOUPLE TRIGONOMENTRY REFERENCE UART WIRELESS BIOMASS CONTACTOR ELECTRIC ILLUMINATION ELECTRICAL SAFETY TRAINING FILTER DESIGN HARDWARE INDUSTRIAL DRIVES JUNCTION FIELD-EFFECT TRANSISTORS NASA NUCLEAR POWER SCIENCE VALVE WWE oscilloscope 3D TECHNOLOGIES COLOR CODES ELECTRIC TRACTION FEATURED FLEXIBLE ELECTRONICS FLUKE GEARMOTORS INTRODUCTION LASSER MATERIAL PID PUMP SEAL ELECTRICIAN CAREER ELECTRICITY SUPPLY AND DISTRIBUTION MUSIC NEUTRAL PERIODIC TABLES OF THE ELEMENTS POLYPHASE AC CIRCUITS PROJECTS REATORS SATELLITE STAR DELTA VIBRATION WATERPROOF