EEE 311: Digital Signal Processing I

Transcription

1 EEE 311: Digital Signal Processing I Course Teacher: Dr Newaz Md Syur Rahim Associated Proessor, Dept o EEE, BUET, Dhaka 1000 Syllabus: As mentioned in your course calendar Reerence Books: 1 Digital Signal Processing: Principles, Algorithms, and Applications John G Proakis 2 Digital Signal Processing: A Practical Approach Emmanuel C Ieachor 3 Schaum s Outlines o Digital Signal Processing 4 Modern Digital Signal Processing Roberto Cristi Course Outlines: This course will cover Chapter 1 through 5 o Proakis s and Chapter 5 through 7 o Ieachor s book

2 Signals Systems and Signal Processing A signal is a unction o one or more independent variables that usually represent time and/ or space A signal contains some kind o inormation that can be conveyed, displayed, or manipulated Examples o signals o particular interests are: Speech, which we encounter in telephony, radio, and everyday lie Biomedical signals, such as electrocardiogram Sound and music, such as reproduced by CD player Video and image, which people watch on television Radar signals, which are used to determine the range and bearing o distant targets A system is a practical device that perorms an operation on a signal to modiy the signal or extract additional inormation rom it A system may be electrical, mechanical, thermal, hydraulic or an algorithm By signal processing we mean the type o operations that is perormed by the system to the signal Digital signal processing is concerned with the digital representation o signals and the use o digital processors to analyze, modiy, or extract inormation rom signals The signals used in most DSP are derived rom analog signals which have been sampled at regular intervals and converted into a digital orm DSP is now used in many areas where analog methods were previously used and in applications which are diicult or impossible with analog method Advantages o DSP The main attractions o DSP are due to the ollowing advantages: Digital signal can withstand channel noise and distortion much better than analog signal Repeaters can be used or long distance digital communication Digital system can be easily modiied with sotware that implements the speciic applications Digital signals can be coded to reduce error rate Storage o digital signal is easy and inexpensive and does not deteriorate with age Reproduction o digital messages is extremely reliable without distortion DSP allows sophisticated applications such as speech recognition and image compression to be implemented with low power portable devices The accuracy is only determined by the number o bits used No drit in perormance with temperature or age Linear phase response can be achieved and complex adaptive iltering algorithms can be implemented using DSP techniques DSP designs can be expensive when large bandwidth signals are involved The ADCs/ DACs may not have suicient resolution or wide bandwidth DSP applications In some DSP systems i an insuicient number o bits are used to represent variables serious degradation in system perormance may result Applications o DSP DSP has revolutionized many areas o science and engineering They are summarized below: Measurements and analysis: Preconditioning the measured signal by rejecting the disturbing noise and intererence The digital ilters can be ound in ECG and EEG equipment to record the weak signals in the presence o heavy background noise and intererence DSP techniques are also used or the analysis o radar and sonar echoes In most GPS receivers today advanced DSP techniques are employed to enhance resolution and reliability [+ patient monitoring, X ray storage, enhancement] Telecommunications: DSP is used in telephone systems or DTMF (dual tone multi requency) signaling, echo cancelling o telephone lines, equalizers or high speed telephone modems, etc Error correcting codes are used to protect digital signals rom bit errors during data trans missions Data compression algorithms are utilized to reduce the number o data bits to represent given inormation DSP is used or speech coding in GSM (global system or mobile communication) telephones, in modulators and demodulators etc [+video conerencing, data communication]

3 Audio and television: Digital signal processing is mandatory in CD players, digital audio tape (DAT) and digital compact cassette (DCC) recorder Digital methods are also used in digital audio broadcasting (DAB) HDTV systems are utilizing lots o digital image processing techniques Digital image processing: Digital image processing is used or restoring blurred or distorted images, data compression, identiication and analysis o pictures and photos [+pattern recognition, satellite weather map, acsimile] Automotive: In automotive business DSP is used or control purposes For example, ignition and injection control system, intelligent suspension system, anti skid brakes, climate control systems, intelligent cruise controllers, airbag controllers etc Some speech recognition and speech synthesis are being tasted in automobiles Experiments have been perormed or background noise cancellation in cars using adaptive digital ilters Basic Elements o DSP Systems The block diagram o atypical DSP system is shown in Figure below The analog input ilter is used to band limit the input signal beore digitization to reduce aliasing The ADC converts the analog input signal into a digital orm The heart o the system is the digital processor (Motorola MC68000, Texas Instruments TMS320C25) The digital processor may implement one o the several DSP algorithms, such as, digital iltering Ater processing the signal may be stored in a computer memory or later use or it may be displayed graphically on a display unit Sampling Sampling is the acquisition o a continuous signal at discrete time intervals The sampled signal is continuous in amplitude but deined only in discrete points in time The process is shown in Figure above The signal obtained in this way is called discrete time signal and is represented as x( n ) x( n) = xa ( nt) ; < n < where, T is the sampling period The inverse o it is sampling requency, F [ F = 1/ T] s s Basic signals 1 Unit sample or unit impulse, δ ( n) 1 n = 0 δ ( n) = 0 n 0

4 xn ( ) = 15 δ ( n+ 2) δ( n+ 1) + 12 δ( n) 05 δ( n 2) + 16 δ ( n 3) Note: Any DT signal can be expanded into, x( n) = x( k) δ ( n k) 2 Unit step, un ( ) k = 1 n 0 un ( ) = 0 n < 0 3 Sinusoidal signals A continuous time sinusoidal signal is deined as, xt () = Acos( Ω 0t+ θ ) A discrete time sinusoid is obtained by sampling a continuous time sinusoid with sampling interval, Ts as, xn ( ) = xnt ( s) = Acos( Ω 0nTs + θ ) = Acos( ω0n+ θ) 2π F0 where, Ω 0Ts = = 2π 0 = ω0is called the digital requency Fs n n 4 Exponential signal, a (or e λ where, a = e λ and λ = α + jβ ) Some peculiarities o discrete time sinusoids There are two unexpected properties o discrete time sinusoids which distinguish them with continuous time sinusoids 1 A continuous time sinusoid is always periodic regardless o its requency, Ω But a Discrete time sinusoid is periodic only i ω is 2π times some rational number 2 A discrete time sinusoid does not have unique waveorm or each value oω In act, discrete time sinusoids with requencies separated by the multiples o 2π are identical Thus a sinusoid cosω0n= cos( ω0 + 2 πk) n= cosω k nwhere k is an integer A discrete time sinusoid xn ( ) = Acos( ω0n+ θ ) is periodic with period N 0, i x( n) = x( n+ N0) Applying this condition m we get, ω0n0 = 2π m or, N0 = 2π N0 and m are integers ω 0

5 π 4π Figure above shows three sinusoids, cos n,cos n and cos 08n The period o irst and the second sinusoids are and 17 respectively The third sinusoid is not periodic From the second property it can be said that sinusoidal signal has unique waveorm over a range o 2π We may select this range to be π to π, 0 to 2π, π to3π etc We shall select this range as π to π We call this range as the undamental range o requencies Thus a sinusoid o any requency ω is identical to some sinusoid o requency ω in the undamental range π to π Thus, cos(87 π n+ θ) = cos(07 πn+ θ) and cos(96 π n+ θ) = cos( 04 πn+ θ) Thereore, the requency 87π is identical to the requency 07π in the undamental range Also the requency 96π is identical to the requency 04π in the undamental range Further reduction in requency range Consider, cos(96 π n+ θ) = cos( 04 πn+ θ) = cos(04 πn θ) This result shows that a sinusoid o any requency ω can always be expressed as a sinusoid o requency ω lies in the requency range 0 to π A systematic procedure to reduce the requency o a sinusoid cos( ωn + θ ) is to express ω as, ω = ω + 2πm ; ω π and m is an integer ω, where Non uniqueness o discrete time sinusoid Figure below shows how two dierent continuous time sinusoids o dierent requencies generate identical discretetime sinusoid

6 Highest oscillation rate in discrete time sinusoid The rate o oscillation o a sinusoid increases continuously as ω increases rom 0 to π The rate o oscillation decreases π 15π ω0 = or 8 8 π 7π ω0 = or 4 4 π 3π ω0 = or 2 2 Asω increases rom π to 2π This is illustrated in Figure above A requency ( π + x) actually appears as the requency ( π x) Sampling continuous time sinusoid and aliasing I two sequences x1( n) = Acos( ω1n+ α1) and x2( n) = Acos( ω2n+ α2) have requencies and phases related by, ω2 = ω1+ 2 kπ, α2 = α1 or, ω2 = ω1+ 2 kπ, α2 = α1 with k an integer, then the two sinusoidal sequences have the same samples, ie x1( n) = x2( n) This is illustrated in Figure below Here, ω 1, ω1 + 2π, ω1 and ω1 + 2π represents the same signal in the time domain I we limit the digital requencyω within the interval π to π then there is one to one correspondence between the signals and their requency representation For each requency in the interval π to π the corresponding aliases are all outside the interval π to π itsel Now, the range o unique digital requencies, π ω π π ΩT π Or, π / T Ω π / T or, π Fs Ω π Fs Ωs Ωs Or, Ω 2 2 This implies that the highest requency o an analog signal must be less than hal the sampling requency to avoid aliasing