DSP First, 2/e. This Lecture: LECTURE #1 Sinusoids. Appendix B: MATLAB
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1 DSP First, 2/e LECURE #1 Sinusoids READING ASSIGNMENS his Lecture: Chapter 2, Sections 2-1 and 2-2 Chapter 1: Introduction Appendix B: MALAB Review Appendix A on Complex Numbers Aug , JH McClellan & RW Schafer 3 CONVERGING FIELDS COURSE OBJECIVE Math Computer Science EE CmpE Physics Applications Students will be able to: Understand mathematical descriptions of signal processing algorithms and express those algorithms as computer implementations (MALAB What are your objectives? BIO Aug , JH McClellan & RW Schafer 4 Aug , JH McClellan & RW Schafer 5
2 WHY USE DSP? Mathematical abstractions lead to generalization and discovery of new processing techniques Computer implementations are flexible Applications provide a physical context Fourier Everywhere elecommunications Sound & Music CDROM, Digital Video Fourier Optics X-ray Crystallography y Protein Structure & DNA Computerized omography Nuclear Magnetic Resonance: MRI Radioastronomy Ref: Prestini, he Evolution of Applied Harmonic Analysis Aug , JH McClellan & RW Schafer 6 Aug , JH McClellan & RW Schafer 7 LECURE OBJECIVES Write general formula for a sinusoidal waveform, or signal From the formula, plot the sinusoid versus time What s a signal? It s a function of time, x(t in the mathematical atca sense UNING FORK EXAMPLE CD-ROM demo A is at 440 Hertz (Hz Waveform is a SINUSOIDAL SIGNAL Computer plot looks like a sine wave his should be the mathematical formula: Acos( 2 (440 t Aug , JH McClellan & RW Schafer 8 Aug , JH McClellan & RW Schafer 9
3 UNING FORK A-440 Waveform ime (sec f 1/ 1000 / Hz ms Aug , JH McClellan & RW Schafer 10 SPEECH EXAMPLE More complicated signal (BA.WAV Waveform x(t ( is NO a Sinusoid heory will tell us x(t is approximately a sum of sinusoids FOURIER ANALYSIS Break x(t into its sinusoidal components Called the FREQUENCY SPECRUM Aug , JH McClellan & RW Schafer 11 Speech Signal: BA DIGIIZE the WAVEFORM Nearly Periodic in Vowel Region Period is (Approximately = sec x[n] is a SAMPLED SINUSOID A list of numbers stored in memory Sample at 11,025 samples per second Called the SAMPLING RAE of the A/D ime between samples is 1/11025 = 90.7 microsec Output via D/A hardware (at F samp Aug , JH McClellan & RW Schafer 12 Aug , JH McClellan & RW Schafer 13
4 SORING DIGIAL SOUND x[n] is a SAMPLED SINUSOID A list of numbers stored in memory CD rate is 44,100 samples per second 16-bit samples Stereo uses 2 channels Number of bytes for 1 minute is 2 X (16/8 X 60 X = Mbytes SINES and COSINES Always use the COSINE FORM Acos( 2 (440 t Sine is a special case: sin( t cos( t 2 Aug , JH McClellan & RW Schafer 14 Aug , JH McClellan & RW Schafer 15 SINUSOIDAL SIGNAL A FREQUENCY Radians/sec Hertz (cycles/sec ( 2 f PERIOD (in sec 1 2 f cos( t AMPLIUDE Magnitude PHASE A Aug , JH McClellan & RW Schafer 16 EXAMPLE of SINUSOID Given the Formula Make a plot 5cos(0.3 t 1.2 Aug , JH McClellan & RW Schafer 17
5 PLO COSINE SIGNAL 5cos( 0. 3 t 12. Formula defines A,, and A Aug , JH McClellan & RW Schafer 18 PLOING COSINE SIGNAL from the FORMULA 5 cos(0.3 t 1.2 Determine period: 2 / 2 / 0.3 Determine a peak location by solving ( t 0 (0.3 t 1.2 Zero crossing is /4 before or after 20 / 3 Positive & Negative peaks spaced by /2 Aug , JH McClellan & RW Schafer 19 0 PLO the SINUSOID 5cos(0.3 t 1.2 Use =20/3 and the peak location at t= Aug , JH McClellan & RW Schafer 20
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