Lecture map 1 to 8 - EC320 Digital Signal Processing(August to November 2009 Lecture map for classes 1 to 8 Instructor Dr K Karthik Lecture 1

Lecture map 1 to 8 - EC320 Digital Signal Processing(August...

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EC320 Digital Signal Processing (August to November 2009) Lecture map for classes 1 to 8 Instructor: Dr. K. Karthik Lecture 1: Introduction to Digital Signal processing Defining signals as a manifestation of some complex process. Signals cannot exist in isolation and emerge from complex systems. Example 1: Speech generation process Example 2: Production of seismic waves – Arriving seismic waves can be used to characterize the behaviour of the underlying system (localize the fault and track system behaviour as it traverses through various states). Applications (two signal-system models): (a) Isolated system model – Earthquake detection, Astronomy. (b) Input-Output model – Channel estimation (Noise + ISI problem), Speaker recognition, Locating and drilling towards oil wells. Other applications: Signal compression (Fourier series representation). Classification of signals: (a) Continuous vs Discrete, (b) Periodic vs Aperiodic, (c) Deterministic vs Random. Examples of random signals. Speech utterance if unpredictable can be treated as a random signal. How does one measure randomness? Presence of noise is an indication of randomness. What is this property which noise possesses? “Entropy”. Lecture 2: Derivation of Continuous time Fourier transform (CTFT) for aperiodic energy signals. Representation of periodic signals. Characterization in terms of Fourier series with convergence criteria (De-Richlets conditions). Fourier series of a periodic rectangular pulse waveform. What happens to the spectrum as the time period increases towards infinity? --- The discrete spectrum becomes continuous and the amplitudes of the sinusoids shrink to zero. However the product of the time period T and the Fourier coefficient k a remains finite. This is nothing but the Fourier transform. Derivation of the Fourier transform starting with the Fourier series of the rectangular pulse. ( ) −∞ = ∆Ω ∆Ω ∆Ω ∆Ω = k t jk T e k R t r π 2 lim ) ( lim 0 ( ) = = d e j R t r t r t j a T π 2 1 ) ( ) ( lim Definition and properties of the impulse function ) ( t δ . Approximation of the delta function in terms of the rectangular pulse and the triangular function. Differentiation of the unit step function. Convergence and properties of the CTFT: (a) Transform of impulse/delayed impulse, (b) Convolution, (c) Modulation, (d) Parseval’s theorem, (e) Sinusoid, (f) Transform domain symmetry properties of a real signal ) ( t x , (g) Transform of derivative of ) ( t x . Lecture 3: Sampling theorem and derivation of the Discrete time Fourier transform from the CTFT Shannon-Nyquist sampling theorem ( B F s 2 > ). Sampling and reconstruction of a bandlimited energy signal ) ( t x a . ( ) ) ( ) ( 2 ) ( ) ( ˆ s s k s a a kT t kT t B Sin kT x t x = −∞ = π π Relating the DTFT to the CTFT. Finding a transform domain representation for the samples )} ( { s a nT x .
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