By Zahir M. Hussain, Amin Z. Sadik, Peter O'Shea
In 3 elements, this booklet contributes to the development of engineering schooling and that serves as a basic reference on electronic sign processing. half I offers the fundamentals of analog and electronic indications and platforms within the time and frequency area. It covers the middle subject matters: convolution, transforms, filters, and random sign research. It additionally treats vital functions together with sign detection in noise, radar variety estimation for airborne objectives, binary verbal exchange platforms, channel estimation, banking and fiscal purposes, and audio results creation. half II considers chosen sign processing structures and methods. middle themes coated are the Hilbert transformer, binary sign transmission, phase-locked loops, sigma-delta modulation, noise shaping, quantization, adaptive filters, and non-stationary sign research. half III offers a few chosen complicated DSP topics.
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Additional info for Digital Signal Processing: An Introduction with MATLAB and Applications
2 5 Time, sec 0 −5 0 5 Frequency, Hz ∠X(f ) 2 0 −2 −10 −5 0 5 10 Frequency, Hz Fig. 12 A decaying exponential signal with its magnitude and phase spectra x(t ) X(f ) 1 1 |X(f )| 1 1/T 2/T 0 −T/2 t 0 T/2 f 0 1/T 3/T 2/T f Fig. 13 A rectangular time pulse with its spectra Example 2 Consider the rectangular time pulse x(t) = PT(t). Its FT is obtained as: Xðf Þ ¼ ZT=2 eÀj2pft dt ¼ Ã 1 Â Àj2pft ÃT=2 1 Â ÀjpfT e e ¼ À ejpfT ÀT=2 Àj2pf Àj2pf ÀT=2 1 ejpfT À eÀjpfT sinðpfTÞ ¼ : ðUsing Euler0 s FormulaÞ 2j pf pf sinðpfTÞ ¼T ¼ TsincðpfTÞ: pfT ¼ The signal, along with its FT and magnitude spectrum, are shown in Fig.
21 (right). 7. In the time domain, power is given by: RT P ¼ T1o 0 o jxðtÞj2 dt. Using the well-known result for the power in a sinusoidal signal yields: 58 W. In the frequency domain one can apply Parseval’s P = 102/2 ? 42/2 = P theorem to get: P = X2k = 22 ? 52 ? 52 ? 22 = 58 W. Example 2 A current signal x(t) flowing through 1X resistor has the form x(t) = e-t/10u(t) A. 1. Determine whether x(t) is a power or energy signal. 2. Find the Fourier transform of x(t). 3. Find the ESD of x(t). Solution: R1 R1 1.
2 Signals in Gaussian Noise If s(t) is a deterministic signal and n(t) is noise, then z(t) = s(t) ? n(t) is a random signal. Consider now the case where s(t) = a (a constant). If n(t) is Gaussian noise with zero mean and variance = r2, then the random variable z(t) is also Gaussian with mean and variance given at any time by: "z ¼ mz ¼ EfzðtÞg ¼ EfðsðtÞ þ nðtÞÞg ¼ Efða þ nðtÞÞg ¼ Efag þ EfnðtÞg ¼ a þ 0 ¼ a; 40 1 Analog Signals and Systems and n o n o varðzÞ ¼ E ðzðtÞ À mz Þ2 ¼ E nðtÞ2 ¼ r2 : Hence, the pdf of the signal z(t) at any time t is given by: 1 zÀa 2 1 pðzÞ ¼ pﬃﬃﬃﬃﬃﬃ e2ð r Þ r 2p While the above result was derived above for the case where s(t) is a constant, its general form is actually valid for any time signal s(t).
Digital Signal Processing: An Introduction with MATLAB and Applications by Zahir M. Hussain, Amin Z. Sadik, Peter O'Shea