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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.

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?

Unit sample sequence:

δ(n) = 1, n = 0 = 0, n ≠ 0

Basic Signals

Unit step sequence:

u(n) = 1, n ≥ 0 = 0, n < 0

Basic Signals

Real-valued exponential sequence:

x(n) = an, a is a real number

Basic Signals

Complex-valued exponential sequence:

x(n) = e(σ + j ω) n

Basic Signals

Sinusoidal sequence:

X(n) = A cos(ω n + θ)

Basic Signals

Random sequences:

rand(1, N)

Basic Signals

Periodic sequence:

x(n) = x(n+N)the smallest integer N is the fundamental period

Basic Signals

Signal addition:

{x1(n)} + {x2(n)} ={x1(n)+x2(n)}

Basic Operations

Signal multiplication

{x1(n)} × {x2(n)} ={x1(n) × x2(n)}

Basic Operations

Scaling:

α {x(n)} ={α x(n)}

Basic Operations

Shifting:

y(n) = { x(n - k) }y(m + k) = { x(m) }

Basic Operations

Folding:

y(n) = {x(-n)}

Basic Operations

Sample Summation:

x(n1)+…+x(n2) = sum(x(n1:n2))

Basic Operations

Sample products

x(n1) × … × x(n2) = prod(x(n1:n2))

Basic Operations

Signal energy:

|x(n1)|2 + … + |x(n2)|2 = sum(abs(x).^2)

Basic Operations

Signal power:

Average power of a periodic signal with fundamental period N1/N (|x(1)|2 +…+|x(N)|2)

Basic Operations

Unit sample synthesis:

Useful Results

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

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

Correlation of sequences:

rx,y(m) = sum_n (x(n) y(n-m))

Useful Results

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

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

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

Let x(n)=u(n)-u(n-10). Decompose x(n) into even and odd components.

Example 4

y(n) = T[x(n)]

Discrete Systems

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

If y(n) = L[x(n)] then L[x(n-k)]=y(n-k)

Linear time-invariant (LTI) system

Impulse Response

Convolution

{ x(n); nxb ≤ n ≤ nxe } and { h(n); nhb ≤ n ≤ nhe }

nyb = nxb + nhb nye = nxe + nhe

Convolution: Matlab Implementation

Correlation is convolution after folding.

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