Lecture notes 6 – Friday 6/30Summary of last lectureSampling, restoration, aliasing and quantizationSamplingRestorationAliasingQuantizationSPHSC 503 – Speech Signal Processing UW – Summer 2006 Lecture notes 6 – Friday 6/30 Summary of last lecture Spectrogram modification • Applying weighting vector to columns of the spectrogram • Simple way to implement LTI filters • More freedom: applying different weighting vector to each column of the spectrogram • Time-varying filter, something that can’t be done by LTI system Reconstruction from spectrogram • Inverse Fourier transform, undoing the window, assembling short sections into signal • Redundancy in spectrogram: each sample of the input is present in two (at 50% window overlap) or four (at 75% window overlap) columns of the spectrogram • Columns of the spectrogram ‘agree’ on the value of that sample • Arbitrary modifications break agreement between columns • Still possible to reconstruct a signal, but reconstruction is not perfect, but acceptable • Despite its imperfect reconstruction, spectrogram modification widely used for signal processing Application of spectrogram modification: noise suppression using spectral subtraction • Given an estimate of the noise spectrum, subtract noise spectrum from a noisy signal’s spectrogram to reduce the noise. • Advanced technique: automatically estimate noise spectrum from noisy signal Sampling, restoration, aliasing and quantization So far in this course we’ve focused on digital signals and systems. But to apply digital systems to real world signals, we need to add the appropriate conversion steps before and after our digital processing. The following diagram gives an overview: The analog signal x(t) is converted by an analog-to-digital (A/D) converter to the sequence x[n], which is processed by an LTI system with output sequence y[n]. The output sequence y[n] is then converted by an digital-to-analog (D/A) converter to an analog signal y(t) The task of the A/D converter is two-fold. First, it must sample the analog signal at a regular rate. This step if often treated as an independent conversion step called continuous-time to discrete-time conversion or C/D conversion. Second, the A/D converter must quantize the samples to a finite number of signal levels that can be represented on the digital platform on which the LTI system is implemented. The task of the D/A converter is to convert the samples back to an analog, continuous-time signal. The A/D converter (including its C/D converter) and the D/A converter must perform their tasks in such a way that they are transparent to the LTI system. That means the entire system, indicated by the dashed box in the diagram above, should have a frequency response that is identical to the frequency response of the LTI system and not be affected by the A/D and D/A conversion. x[n] LTI system y[n] x(t) y(t) A/D D/A – 1 –SPHSC 503 – Speech Signal Processing UW – Summer 2006 Sampling The first conversion performed by an A/D converter is sampling. In this conversion step, a continuous-time signal is converted to a discrete-time signal. A discrete-time signal is a signal that is sampled, but that still can take on any signal value at its sample points. In contrast, a digital signal is a sampled signal that is quantized and can only take on a finite set of discrete signal values at its sample points. The sampling conversion step is also called C/D conversion. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1-0.500.51Time (s)AmplitudeContinuous-time signal 0 10 20 30 40 50 60 70 80 90 100-1-0.500.51Index (n)AmplitudeDiscrete-time signal Theoretically, the sampling or C/D conversion consists of two steps: • Keep the signal values at the sampling times, and set the signal to zero everywhere else • Normalize the time axis from seconds to sample index The first step can be viewed as multiplying an analog signal x(t) by an analog impulse train s(t), resulting in a sampled analog signal ()sxt , as illustrated by the figure below: 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1-0.500.51Time (s)AmplitudeAnalog signal, x(t) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1-0.500.51Time (s)AmplitudeAnalog impulse train, s(t) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1-0.500.51Time (s)Amplitudexs(t) = x(t) ⋅ s(t) The spacing of the impulses of the impulse train is called the sampling period, specified by the symbol T, and is the reciprocal of the sampling frequency, Fs, i.e., T = 1 / Fs. Multiplying the signal with an impulse train in the time-domain is the same as convolving the spectrum of the signal with the spectrum of the impulse train. We saw this ‘multiplication in time is convolution in frequency’ property of the Fourier transform before when we discussed multiplying a sequence with a window to obtain its short-term spectrum. To understand the effect of multiplication here, we need to know the Fourier transform of the impulse train. Without proof, – 2 –SPHSC 503 – Speech Signal Processing UW – Summer 2006 we will state that the Fourier transform of an impulse train in time is an impulse train in frequency. The impulse trains in time and frequency are related according to the figure below. A spacing of T = 1 / Fs seconds of impulses with an amplitude of 1 in the time domain results in a spacing of Fs Hz of impulses of amplitude Fs in the frequency domain. Therefore, given the spectrum of an analog signal, we can find the spectrum of the sampled analog signal by convolving it with the spectrum of the impulse train: The signal’s spectrum is shifted and weighted by the impulses in the spectrum of the impulse train signal. As a result, the sampled analog signal’s spectrum is periodic, and its period is equal to the sampling frequency. The next step in the C/D converter is normalization of the time-axis. Before normalization, the sampled analog signal xs(t) has non-zero values at t = …,-2T,-T,0,T,2T,…. To normalize these sampling times, we divide t by the sampling period T, such that the signal now has non-zero values at t = …-2,-1,0,1,2,… . The signal values at these time instances are then treated at the values of the sequence x[n] at the corresponding index. As a result, the relationship between the analog signal x(t) and the sequence x[n] is[] ( )xnxnT= . Normalization of the time axis by T in the time domain causes a normalization of the frequency axis by Fs in