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Digital Signal Processing (DSP) Systems

Digital Signal Processing (DSP) Systems . Digital processing of analog signals (mixed signal applications) forms one of the most important applications of DSP theory. ...101011. …001010. A/D converter. DSP system. D/A converter. analog input. digital input. digital output.

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Digital Signal Processing (DSP) Systems

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  1. Digital Signal Processing (DSP) Systems • Digital processing of analog signals (mixed signal applications) forms one of the most important applications of DSP theory. ...101011... …001010... A/D converter DSP system D/A converter analog input digital input digital output analog output sampling and quantization discrete to continuous antialiasing prefilter reconstruction filter EE421, Lecture 1

  2. Spectral Representation of Continuous-Time Signals • Fourier Analysis: • f represents frequency in units of cycles/second or Hz and W represents frequency in units of radians/second. • If x(t) is a voltage signal, then X(W) and X(f) have units of volts/Hz. • The conversion between frequency variables is W = 2p f. • Fourier Synthesis: or or EE421, Lecture 1

  3. Some Important Fourier Pairs • Constant (DC) signal: (this signal contains only DC or zero frequency) • Impulse: (this signal contains all frequencies) • Complex exponential (sinusoid): (this signal contains only one frequency component - in fact, this signal is used to define frequency!) • Real sinusoid: (this signal contains two frequency components, +/- f0)These pairs are for frequency measured in Hz. Remember the following rule for changing variables with impulses: EE421, Lecture 1

  4. Signals are “Sums of Sinusoids” • Periodic signals contain only discrete frequency components that are multiples of the fundamental frequency. • Non-periodic signals contain a continuous set of frequency components. The amplitude and phase of each sinusoid forms the spectrum of the signal! EE421, Lecture 1

  5. Linear, Time-Invariant (LTI) Systems • Impulse response: • Convolution: • Frequency Response: LTI System 0 t 0 t LTI System LTI System EE421, Lecture 1

  6. Filtering • H(f) modifies the amplitude of the input signal’s spectrum according to : • H(f) modifies the phase of the input signal’s spectrum according to: LTI System EE421, Lecture 1

  7. Review of Sampling • A bandlimited signal is one whose frequency spectrum contains no components greater than some maximum frequency fmax. • The sampling theorem states that bandlimited signals can be reconstructed perfectly from their samples provided the sampling rate fs (in samples/second) satisfies:2 fmax is called the Nyquist rate. X(f) X(f) -fmax 0 fmax f 0 f non-bandlimited signal bandlimited signal EE421, Lecture 1

  8. -fs -fs 0 0 fs fs f (Hz) f (Hz) Aliasing • Sampling at a rate slower than the Nyquist rate will result in aliasing. That is, frequency components greater than fs / 2 will be folded back into the Nyquist interval. This is generally a bad thing. Don’t let this happen to you! EE421, Lecture 1

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