Down sampling discarding samples decimation down

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Down-sampling = discarding samples = decimation Down Sampling In down-sampling by an integer factor M > 1, every Mth samples of the input sequence are kept and M - 1 in-between samples are removed Dr Sajjad Zaidi - DSP 19
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Operation on Signals Up Sampling Up-sampling is the converse of downsampling L-1 zero values are inserted between each pair of original values. Dr Sajjad Zaidi - DSP 20
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Complex Numbers Complex numbers are a mathematical convenience that lead to simple expressions. A 2 nd “imaginary” dimension (j≡√ -1) is added to all values. Rectangular Form: x = x re + j·x im where magnitude and phase θ Polar Form When adding, real and imaginary parts add: (a+jb) + (c+jd) = (a+c) + j(b+d) When multiplying, magnitudes multiply phases add: Dr Sajjad Zaidi - DSP 21
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Complex Numbers Complex Conjugate Flips imaginary part / negates phase Useful in resolving to real quantities: Addition Multiplication Dr Sajjad Zaidi - DSP 22
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Classes of Sequences Finite/infinite (extent in n) Real/complex: x[n] = x re [n] + j·xi m [n] wrt to Symmetry Conjugate symmetric sequence Conjugate antisymmetric: Dr Sajjad Zaidi - DSP 23
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Conjugate symmetric decomposition Any sequence can be expressed as conjugate symmetric (CS) / antisymmetric (CA) parts: x[n] = x cs [n] + x ca [n] where: x cs [n] = 1/2(x[n] + x*[-n]) = x cs *[-n] x ca [n] = 1/2(x[n] x*[-n]) = -x ca *[-n] When signals are real, CS =Even (x re [n] = x re [-n]), CA = Odd (x re [n] = -x re [-n]), Dr Sajjad Zaidi - DSP 24
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Dr Sajjad Zaidi - DSP 25 Basic Sequences
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Unit Sample Sequence It is defined as Shift in time Can express any sequence with δ Dr Sajjad Zaidi - DSP 26
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Unit step sequence It is defined as Related to unit sample: Dr Sajjad Zaidi - DSP 27
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