Review of Fourier Transform for Continuous-time aperiodic signals.

Summary of the approach to obtain the Fourier Transform for Discrete-time aperiodic signals.

Development of Discrete-time Fourier Transforms. DTFT. x[n} -N1 N2 -N -N1 N2 N

Development of Discrete-time Fourier Transforms.

Development of Discrete-time Fourier Transforms.

Development of Discrete-time Fourier Transforms.

Development of Discrete-time Fourier Transforms.

Discrete-time Fourier Transform

Example 5.1 Discrete-time Fourier Transform.

Example 5.1 Discrete-time Fourier Transform. |X(w)| w -p 2p p Phase X(w) -p p 2p

Discrete-time Fourier Transform for rectangular pulse Example 5.3 | 1 x[n] -2 2 n X(w) 5 w -p p

X(w) w -p p

Review of relationships for Fourier transform associated with continuous-time periodic signals

Summary of relationships for DT Fourier transform associated with DT periodic sequences

-N -N1 N2 N

Illustration of relationship of DTFS & DTFT -N1 N1 N n Case N=20, 2N1+1=5. 1/4 1/4 | 5 6 7 8 12 1 2 3 4 9 10 11 -1 19 20 -2 x2p/20

Summary of relationships for DT Fourier transform associated with DT periodic sequences

Discrete-time Fourier Transforms. DTFT. The synthesis equation for the discrete sequence x[n] (which is aperiodic in nature ) is made up of the linear combination of the basic complex exponential signal. The combination here is through the weights of X(jw) and the integral combination over an interval period of 2 pi radians. The analysis equation is the equation that covert the sequence in the time domain to the frequency. Notice that although the time domain signal or sequence is discrete in nature, the frequency domain is not discrete but tends to be continuous in frequencies. The spectrum X(w) is complex and it has the real part and the imaginary part. Or in the polar coordinate form it has the magnitude and the phase/angle.

Properties of Discrete-time Fourier Transforms.

Properties of Discrete-time Fourier Transforms.

Properties of Discrete-time Fourier Transforms.

Properties of Discrete-time Fourier Transforms.

Convolution Property x[n] h[n] H(w) h[n]*x[n] X(w) H(w)X(w)

Filtering Changing the relative amplitudes of the frequency components. Linear time-invariant systems that change the shape of the spectrum are referred to as frequency-shaping filters. Systems that pass some frequencies and attenuate or eliminate others are referred to as frequency-selective filters.

Frequency-shaping Filters Audio Systems LTI filters allows the listeners to modify the relative amounts of low-frequency energy(bass) and high frequency energy (treble). Frequency response of these filters are changed via manipulating the tone control of your Hi-Fi system.

Frequency-shaping Filters Also in Hi-Fi Audio Systems we will have equalizing filters included in the preamplifier to compensate for the frequency response characteristics of the speakers. Overall these cascaded filtering stages are frequently referred to as the equalizing or equalizer circuits for the audio system.

Magnitude Frequency Plot of the Frequency response of LTI Systems. Bode Plot 20log10|H(w)| in decibels (dB) 0db 100 1000 10 Frequency in logarithmic scale

Phase Frequency Plot of the Frequency response of LTI Systems. Bode Plot Angle H(jw) P/2 10 100 1000 -p/2 Frequency in logarithmic scale

Magnitude Frequency Plot of the Frequency response of Differentiating Filters. Enhancing rapid variations or transitions in a signal e.g. enhance edges in picture processing. |H(w)| y(t)=dx(t)/dt H(w)=Y(w)/X(w)=jw. Frequency

Frequency-Selective Filters Frequency response of a discrete-time ideal lowpass filter H1(w) p 2p -2p -p -wc wc Passband w Stopband Stopband

Frequency response of a discrete-time ideal lowpass & highpass filter H1(w) 2p -2p -p -wc p wc w H2(w) p -p

Frequency response of a discrete-time ideal lowpass & bandpass filter H1(w) -2p -p -wc p 2p wc w H2(w) p -p