EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 08-Sept, 98EE421, Lecture 11 Digital Signal Processing (DSP) Systems l Digital processing of analog signals (mixed signal applications) forms one of the most important applications of DSP theory. A/D converter DSP system D/A converter … analog input digital input analog output digital output antialiasing prefilter sampling and quantization discrete to continuous reconstruction filter

EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 08-Sept, 98EE421, Lecture 12 Spectral Representation of Continuous-Time Signals l Fourier Analysis: –f represents frequency in units of cycles/second or Hz and  represents frequency in units of radians/second. –If x(t) is a voltage signal, then X(  and X(f) have units of volts/Hz. –The conversion between frequency variables is  = 2  f. l Fourier Synthesis: or

EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 08-Sept, 98EE421, Lecture 13 Some Important Fourier Pairs l Constant (DC) signal: ( this signal contains only DC or zero frequency ) l Impulse: ( this signal contains all frequencies ) l 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, +/- f 0 ) These pairs are for frequency measured in Hz. Remember the following rule for changing variables with impulses:

EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 08-Sept, 98EE421, Lecture 14 Signals are “Sums of Sinusoids” l Periodic signals contain only discrete frequency components that are multiples of the fundamental frequency. l Non-periodic signals contain a continuous set of frequency components. The amplitude and phase of each sinusoid forms the spectrum of the signal!

EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 08-Sept, 98EE421, Lecture 15 Linear, Time-Invariant (LTI) Systems l Impulse response: l Convolution: l Frequency Response: t t 0 0

EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 08-Sept, 98EE421, Lecture 16 Filtering l H(f) modifies the amplitude of the input signal’s spectrum according to : l H(f) modifies the phase of the input signal’s spectrum according to:

EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 08-Sept, 98EE421, Lecture 17 Review of Sampling A bandlimited signal is one whose frequency spectrum contains no components greater than some maximum frequency f max. The sampling theorem states that bandlimited signals can be reconstructed perfectly from their samples provided the sampling rate f s (in samples/second) satisfies: 2 f max is called the Nyquist rate. 0f max -f max ff0 X(f)

EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 08-Sept, 98EE421, Lecture 18 Aliasing Sampling at a rate slower than the Nyquist rate will result in aliasing. That is, frequency components greater than f s / 2 will be folded back into the Nyquist interval. This is generally a bad thing. 0fsfs -f s f (Hz) 0fsfs -f s f (Hz) Don’t let this happen to you!