Digital signal processing
[...]... eliminate frequency components in the signal above the Nyquist limit lowpass filter output gain ws ws frequency rad/s 2 Figure 2.4 Lowpass digital filter magnitude spectrum 15 Digital Signal Processing lowpass filter output gain Figure 2.5 2.4 frequency rad/s ws ws 2 Lowpass digital filter showing Nyquist violated Anti-aliasing Filters Because the frequency content of most signals is unknown to some degree,... sampled signal frequency response as a discrete time signal is periodic and symmetric 19 Digital Signal Processing every ws rad/s or 1/T Hz and looks like Figure 2.3 for any w, where w1 is w + ws and w2 is –w + ws Summary In this chapter the effect of periodic sampling of an analog signal is shown to generate two important characteristics These characteristics were developed by using a test signal composed... = 2cos(72.83t) 4 For a digital filter system with the given ADC sample periods T , compute the Nyquist limit (a) T = 0.1 s (b) T = 0.002 s 5 Determine which input signals to a digital filter or DSP system will be aliased by the given sample period T (a) x(t) = 2cos(10t), T = 0.1 s (b) x(t) = 8cos(15t), T = 0.2 s 21 Digital Signal Processing 6 Determine whether the following signals will be aliased... equation of any signal after sampling, not just that of the cosine signal used here For example, the following equations are the input and sampled output signals of an ADC for a decaying sinusoidal signal x(t) = Ae–3t cos(7t) x(n) = x(nT ) = Ae–3nT cos(7nT ) Figure 2.1 10 ADC samples at nT = n(0.1) for slow cosine input Effect of Signal Sampling Table 2.1 Showing the effects of sampling the signals x(t),... x(10) = –1.877 3 x(0) = 2.0, x(1) = 1.08 23 Digital Signal Processing 4a 31.4 rad/s 4b 1570.7 rad/s 5a not aliased 5b aliased 6a not aliased 6b aliased, –3sin(4.89t) 6c aliased, 5cos(154.2t) 7 24 x(n) = 6cos(0.7n) Digital Filter Specification s c h a p t e r 3 Digital Filter Specifications Introduction In this chapter we begin the first step in designing digital filters, which is drawing their graphical... illustrates important ideas about filters The first is that digital filter designers, as opposed to digital control designers, are usually interested only in sinusoidal signals The second is that digital filter designers are usually interested only in the amplitudes of sinusoidal signals and not in the phase (the phase is extremely significant for digital control) Finally, the gain is a ratio of output... as shown in the following equation 27 Digital Signal Processing loss = (gain)–1 and lossdB = –gaindB 3.2 The Lowpass Digital Filter Specification One of the easiest graphical specifications to draw is that for the lowpass digital filter From analog filtering the student remembers that a lowpass filter is supposed to pass (or not reduce very much) low frequency signals, while it should stop (or reduce... function is identical if its angle is changed by 2π radians, we will show that the signal x2(t) in Equation 2.4 has the same value out of the ADC as the original signal x(t) x2(t) = cos[(–w + ws)t ] (Equation 2.4) This cosine signal x2(t) is just the original signal x(t) with its frequency at the sample frequency minus the original signal s frequency When samples of x2(t) are taken every T seconds by the ADC,... – 5cos(40t) 22 Effect of Signal Sampling 4 For a digital filter system with the given ADC sample periods T , compute the Nyquist limit (a) T = 0.025 s (b) T = 001 s 5 Determine which input signals to a digital filter or DSP system will be aliased by the given sample period T (a) x(t) = –2cos(10t), T = 0.3 s (b) x(t) = 4sin(105t), T = 0.03 s 6 Determine whether the following signals will be aliased... 2.3) The sampling frequency in Hz is 1/T and in rad/s is 2π/T , so Equation 2.3 gives x1(n) = cos[(w+2π/T )nT ] = cos(wnT + 2nπ) = cos(wnT ) 11 Digital Signal Processing Notice that after sampling, the signal in Equation 2.3 looks just like the original sampled signal in Equation 2.2 You can see that the sampled values of Equation 2.3 shown in column 4 of Table 2.1 are the same as the values in column . done to a signal using coding on a computer or DSP chip. This includes digital filtering 2 Digital Signal Processing of signals as well as digital integration and digital correlation of signals. This. done only on analog or continuous time signals using analog signal processing (ASP). Until the late 1950s digital Introduction to Digital Signal Processing and Digital Filtering chapter 1 1 computers. replace analog systems. For digital filtering, these processing methods are discussed in Chapters 10 and 11. 1.3 Simple Examples of Digital Signal Processing Digital signal processing entails anything