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Note for Digital Signal Processing - DSP by CHEVELLA ANILKUMAR

  • Digital Signal Processing - DSP
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Figure 1.1: The Unit Impulse Sequence ii) Unit Step Sequence The unit step sequence is one that has amplitude of zero for negative indices and amplitude of one for non-negative indices. 0 𝑛<0 𝑢 𝑛 = 1 𝑛≥0 It is shown in Figure 1.2. Figure 1.2: The Unit Step Sequence iii) Exponential sequences The general form is If A and α are real numbers then the sequence is real. If 0 < α < 1 and A is positive, then the sequence values are positive and decrease with increasing n:

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iv) Sinusoidal Sequences A sinusoidal sequence has the form This function can also be decomposed into its in-phase xi[q] and quadrature xq[n] components. This is a common practice in communications signal processing. It is shown in Figure 1.3. Figure 1.3: The Sinusoidal Sequence v) Complex Exponential Sequences Complex exponential sequences are essentially complex sinusoids. vi) Random Sequences The sample values of a random sequence are randomly drawn from a certain probability distribution. They are also called stochastic sequences. The two most common distributions are the Gaussian (normal) distribution and the uniform distribution. The zero-mean Gaussian distribution is often used to model noise. Figure 1.4 and Figure 1.5 show examples of uniformly distributed and Gaussian distributed random sequences respectively.

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Figure 1.4: Uniformly distributed random sequence with amplitudes between -0.5 and 0.5. Figure 1.5: Gaussian distributed random sequence with zero mean and unit variance. Types of Sequences The discrete-time signals that we encounter can be classified in a number of ways. Some basic classifications that are of interest to us are described below. i) ii) iii) Real vs. Complex Signals A sequence is considered complex at least one sample is complex-valued. Finite vs. Infinite Length Signals Finite length sequences are defined only for a range of indices, say N1 toN2. The length of this finite length sequence is given by |N2- N1 + 1|. Causal Signals A sequence x[n] is a causal sequence if x[n] = 0 for n < 0.

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iv) Symmetric Signals First consider a real-valued sequence {x[n]}. Even symmetry implies that x[n] = x[-n] and Odd symmetry implies that x[n] = -x[n] for all n. Any real-valued sequence can be decomposed into odd and even parts so that where the even part is given by and the odd part is given by A complex-valued sequence is conjugate symmetric if x[n] = x*[-n]. The sequence has conjugate anti-symmetry if x[n] = -x*[-n]. Analogous to real-valued sequences, any complexvalued sequence can be decomposed into its conjugate symmetric and conjugate antisymmetric parts: v) Periodic Signals A discrete-time sequence is periodic with a period of N samples if for all integer values of k. Note that N has to be a positive integer. If a sequence is not periodic, it is aperiodic or non-periodic. Consider a discrete-time sequence x [n] based on a sinusoid with angular frequency ωo: If this sequence is periodic with a period of N samples, then the following must be true: However, the left hand side can be expressed as and the cosine function is periodic with a period of 2π and therefore the right hand side of the above equation is given by for integer values of r. Comparing the above two equations, we have

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