discrete-time signals & systems
DESCRIPTION
Discrete-time Signals & Systems. Discrete-Time Signals. The correct representation of a discrete-time signal in Matlab takes 2 vectors. One vector is used to indicate the locations of the time samples. - PowerPoint PPT PresentationTRANSCRIPT
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Discrete-time Signals & Systems
Discrete-Time Signals
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The correct representation of a discrete-time signal in Matlab takes 2 vectors.
One vector is used to indicate the locations of the time samples.
The other vector is used to indicate the amplitude (value) of the signal at the corresponding temporal locations.
How to represent a discrete-time signal in Matlab?
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Unit sample sequence:
δ(n) = 1, n = 0 = 0, n ≠ 0
Basic Signals
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Unit step sequence:
u(n) = 1, n ≥ 0 = 0, n < 0
Basic Signals
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Real-valued exponential sequence:
x(n) = an, a is a real number
Basic Signals
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Complex-valued exponential sequence:
x(n) = e(σ + j ω) n
Basic Signals
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Sinusoidal sequence:
X(n) = A cos(ω n + θ)
Basic Signals
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Random sequences:
rand(1, N)
Basic Signals
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Periodic sequence:
x(n) = x(n+N)the smallest integer N is the fundamental period
Basic Signals
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Signal addition:
{x1(n)} + {x2(n)} ={x1(n)+x2(n)}
Basic Operations
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Signal multiplication
{x1(n)} × {x2(n)} ={x1(n) × x2(n)}
Basic Operations
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Scaling:
α {x(n)} ={α x(n)}
Basic Operations
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Shifting:
y(n) = { x(n - k) }y(m + k) = { x(m) }
Basic Operations
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Folding:
y(n) = {x(-n)}
Basic Operations
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Sample Summation:
x(n1)+…+x(n2) = sum(x(n1:n2))
Basic Operations
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Sample products
x(n1) × … × x(n2) = prod(x(n1:n2))
Basic Operations
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Signal energy:
|x(n1)|2 + … + |x(n2)|2 = sum(abs(x).^2)
Basic Operations
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Signal power:
Average power of a periodic signal with fundamental period N1/N (|x(1)|2 +…+|x(N)|2)
Basic Operations
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Unit sample synthesis:
Useful Results
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Even and odd synthesis Even signal: xe (-n) = xe (n) Odd signal: xo (-n) = - xo(n) x(n) = xe(n) + xo (n),
xe(n) = ½ (x(n) + x(-n))xo(n) = ½ (x(n) - x(-n))
Useful Results
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The geometric series
1 + α + α2 + … + α∞ = 1/(1-α) for |α| < 1
1 + α + α2 + … + αN-1 = (1-αN)/(1-α)for any α
Useful Results
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Correlation of sequences:
rx,y(m) = sum_n (x(n) y(n-m))
Useful Results
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x(n) = 2δ(n+2) – δ(n-4), -5≤n≤5
x(n)=n[u(n)-u(n-10)]+10e-0.3(n-10)
[u(n-10)-u(n-20)], 0≤n≤20x(n)=cos(0.04πn)+0.2w(n), 0≤n≤50, where w(n) is a Gaussian random sequence with zero mean and unit variance
x(n)={…,5,4,3,2,1,5,4,3,2,1,5,4,3,2,1,…}; -10≤n≤9Example 1
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Let x(n) = {1,2,3,4,5,6,7,6,5,4,3,2,1}. Determine and plot the following sequences.
x1(n)=2x(n-5)-3x(n+4) x2(n)=x(3-n)+x(n)x(n-2)
Example 2
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Generate the complex-valued signal x(n)=e(-0.1+j0.3)n, -10≤n≤10And plot its magnitude, phase, the real part and the imaginary part in four separate subplots.
Example 3
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Let x(n)=u(n)-u(n-10). Decompose x(n) into even and odd components.
Example 4
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y(n) = T[x(n)]
Discrete Systems
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A discrete system L[] is linear, if and only if it satisfies the principle of superposition.
L[a1x1(n)+a2x2(n)]=a1L[x1(n)] + a2L[x2(n)]
Linear Discrete Systems
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If y(n) = L[x(n)] then L[x(n-k)]=y(n-k)
Linear time-invariant (LTI) system
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Impulse Response
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Convolution
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{ x(n); nxb ≤ n ≤ nxe } and { h(n); nhb ≤ n ≤ nhe }
nyb = nxb + nhb nye = nxe + nhe
Convolution: Matlab Implementation
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Correlation is convolution after folding.
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x(n)=[3, 11, 7, 0, -1, 4, 2] y(n)=x(n-2)+w(n), where w(n) is a sequence of random noise
Compute the cross-correlation between y(n) and x(n)
Example 5