Lecture notes 8 – Wednesday 7/5Summary of last lecturePractical quantizationIIR filter designPhase response and group delayButterworth filter plotsChebyshev type I filter plotsChebyshev type II filter plotsElliptic filter plotsSPHSC 503 – Speech Signal Processing UW – Summer 2006 Lecture notes 8 – Wednesday 7/5 Summary of last lecture Quantization • Second step of A/D, representing continuous signal level on a finite number of levels • Introduces quantization error or quantization noise • Spectrum of quantization noise is important: o sample and hold A/D converter has flat quantization noise spectrum o sigma-delta A/D converter has shaped quantization noise spectrum, low for low frequencies, high for high frequencies FIR filter design • Windowing method (fir1): o have analytic expression for ideal frequency response o find analytic expression for ideal impulse response o truncate or window ideal impulse response to desired length • Frequency sampling (fir2 and firls) o have analytic expression for ideal frequency response o sample ideal frequency response at N sample points o take inverse DTFT to find N-point impulse response o optionally, designate frequency samples in transition band as free samples, optimize impulse response that minimizes the difference between ideal and effective frequency response • Equiripple design with the Parks-McClellan algorithm (firpm) o Put ripples uniformly over pass-band and stop-band for lower order o Parks-McClellan algorithm computes optimal impulse response given frequency response Practical quantization Some more research into quantization uncovered some practical information about it: - CD’s use 16-bit quantization (65536 signal levels, good) - Digital telephones use 8-bit quantization (256 signal levels, ok/poor for speech) - Signals generated in Matlab use 64 bits (>18 billion billion signal levels) - Wav-files use 16-bit quantization by default, capable of 32 bits (> 4 billion signal levels) - If a signal uses the maximum range in signal levels (-1 to +1), then its average power level is roughly 0.7, and 16 bit quantization noise is of the order 1610(2/2 )20log 87dB0.7⎛⎞⎟⎜⎟=−⎜⎟⎜⎟⎜⎝⎠ - For some other numbers of bits used in quantization, quantization noise is of the order: 1-bit = 3 dB, 2-bit = -3 dB, 4-bit = -15 dB 8-bit = -39 dB, 16-bit = -87 dB, 32-bit = -184 dB, 64-bit = -376 dB - If a signal doesn’t use the maximum signal level range, then quantization noise becomes more audible – 1 –SPHSC 503 – Speech Signal Processing UW – Summer 2006 IIR filter design Last lecture we saw the design methods for FIR filters. Design of IIR filters works different, and in general takes the following steps: • Given a desired magnitude response • Select an analog prototype low-pass filter for a Butterworth, Chebyshev, or elliptic filter. Definitions of such prototypes are available in the literature. They are usually defined by magnitude-squared response instead of their magnitude response. • Prototypes are only available for low-pass filters with a fixed cut-off frequency. The prototype must go through a frequency band transformation to adjust the cut-off frequency, and to change the type of filter from low-pass to band-pass, high-pass or band-stop depending on the desired filter • The filter is then transformed from the analog domain to the digital domain We will ignore the precise details of these steps, and focus on the characteristics of the various types of IIR filters and the Matlab functions that generate them (according to the steps above). Butterworth (Matlab: buttord and butter) A Butterworth filter is characterized by the property that its magnitude response is flat in both the pass-band and the stop-band, and is monotonically decreasing, i.e., no ripples. As we can see in the magnitude-squared plot of the Butterworth filter, the magnitude response for F = 0 is always 1 for all N. The magnitude response at the cut-off frequency is always ½ for all N. As the order of the filter increases, the Butterworth filter approaches an ideal low-pass filter. Chebyshev Type I (Matlab: cheb1ord and cheby1) and Chebyshev Type II (Matlab: cheb2ord and cheby2) There are two types of Chebyshev filters. The Chebyshev Type I filters have equiripple response in the pass-band, while the Chebyshev Type II filters have equiripple response in the stop-band. Recall our discussion of equiripple FIR filters, where we saw that we can obtain lower order filters that meet our design requirements when we choose a filter that has an equiripple rather than a monotonic behavior. Likewise, Chebyshev filters provide lower order than Butterworth filters for the same specifications. Elliptic (Matlab: ellipord and ellip) Elliptic filters exhibit equiripple behavior in the pass-band as well as in the stop-band. They are similar in magnitude response characteristics to the FIR equiripple filters. Therefore, elliptic filters are optimum filters in that they achieve the minimum order N for the given specifications, or alternatively, achieve the sharpest transition band for a given order N. Phase response and group delay Elliptic filters provide optimal performance in the magnitude-squared response, but have highly non-linear phase response, which is undesirable in many applications. Although we’re mostly concerned with the magnitude response of the filters that we design, phase is still an important issue in the overall system. At the other end of the performance scale are the Butterworth filters, which have maximally flat magnitude response and require a higher order N to achieve the same stop-band specification. However, they exhibit a fairly linear phase response in their pass-band. The Chebyshev filters have phase characteristics that lie somewhere in between. Therefore in practical application you can consider Butterworth as well as Chebyshev filters, in addition to – 2 –SPHSC 503 – Speech Signal Processing UW – Summer 2006 elliptic filters. The choice depends on both the filter order, which influences processing speed and implementation complexity, and the phase characteristics, which control the distortion. When using an IIR filter with a non-linear phase response, it is usually not easy to tell from the phase response how the filter will distort a signal. Another form of filter analysis called group delay gives more insight into the distortion