# 1 Discrete-time signals & Systems - Universitetet i oslo 3.1 Continous-time signals The...

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1 Discrete-time signals & Systems

1.1 Discrete-time signals

Discrete time signal, x[n] = {. . . , x[−1], x ↑ [0], x[1], . . .}.

Unit sample sequence, δ[n].

δ[n] =

{ 1, n = 0

0, otherwise.

Any abritary sequence x[n] can be synthesized as x[n] = ∑∞ k=−∞ x[k]δ[n− k].

Unit step sequence, u[n]

u[n] =

{ 1, n ≥ 0 0, otherwise.

Unit ramp sequence, ur[n]

ur[n] =

{ n, n ≥ 0 0, otherwise.

Exponential sequences x[n] = an ∀n A one-sided exponential sequence αn, n ≥ 0; α ∈ < is called a geometric series.∑∞ n=0 α

n −→ 11−α , |α| < 1.∑N n=0 α

n −→ 1−α N

1−α ,∀α.

Sinusoidal sequences x[n] = cos[w0n+ Φ], ∀n, w0, Φ ∈

Time-invariant versus time-variant systems A linear system T is time-invariant or shift-invariant iff the following is true: x(n) −→ H{·} −→ y[n] −→ Shift by k −→ y[n− k].

x(n) −→ Shift by k −→ x[n− k] −→ H{·} −→ y[n− k]. A linear, time-invariant system is denoted a LTI-system.

LTI-systems and convolution sum Let x[n] and y[n] be the input-output pair of an LTI-system. Then the output is given by: y[n] = H{x[n]} =

∑∞ k=−∞ x[k]H{δ[n− k]} ≡

∑∞ k=−∞ x[k]h[n− k].

h[n] is the respons of an LTI system to δ[n] and called the impulse response. A LTI-system is completey descried in time-domain by the impulse response, h[n]. The mathematical operator y[n] ≡ x[n]∗h[n] =

∑∞ k=−∞ x[k]h[n−k] is called linear convolution

sum.

Static versus dynamic systems A system is static or memoryless if the outpt at any time n = n0 depends only on the input at time n = n0.

Causal versus noncausal systems A system is causal if, for any n0, the system response at time n0 only depends on the input up to time n = n0. A LTI-system is causal iff h[n] = 0, n < 0.

Stable versus unstable systems A system is bounded-input bounded-output (BIBO) stable if, for any input that is bounded, |x[n]| ≤ A

2 The z-transform

2.1 The z-transform

Definition of the two-sided z-transform.

X(z) =

∞∑ n=−∞

x[n]z−n.

Region of convergence (ROC) The region of convergence is the subset of the complex plane where the z-transform converges. For an FIR-filter (or a finite-length signal) it is the entire complex plane with the possible excep- tions of 0 or∞. For an IIR-filter (or infinite-length signal) it is one of three:

1. The outside of a disc |z| > a for a causal system.

2. A disc |z| < a for an anti-causal system.

3. The intersection between the two abovementioned regions a < |z| < b for a two-sided system.

Poles and Zeroes A pole is a point z ∈ C where H(z) =∞. A zero is a point z ∈ C where H(z) = 0.

Relationship between the z-transform and the discrete-time Fourier transform. X(eω) = X(z)|z=ejω

Some common z-transforms. Being able to read and use tables like these:

3

Properties of the z-transform. Being able to prove some of these:

Calculating a transfer function/system function from a difference equation. Calculate the z-transform of the difference equation for y[n] and divide by X(z):

H(z) = Y (z)

X(z) .

Determining and interpreting pole-zero-plots with respect to:

1. Stability.

2. Causality.

3. Symmetry.

4. Real or complex (time domain) signals.

5. The connection to the transfer function.

6. Approximating the frequency response, determining the filter type.

Stability: The ROC contains the unit circle. Causality: The ROC is |z| > α - causal system (h[n] = 0 for n < N ). The ROC is |z| < α - anti-causal system (h[n] = 0 for n > N ). The ROC is α < |z| < |β| - two-sided system (there is no interval towards pos. or neg. infinity in which h[n] is zero). Symmetry: The system/signal is symmetric in the time domain iff poles and zeroes come in reciprocal pairs. Meaning: z is a pole of H(z)⇔ z−1 is a pole of H(z). (The same applies for zeros.) Real or complex time domain signal: The signal is real in the time domain iff all poles and zeros come in complex conjugate pairs. Meaning: z is a pole of H(z)⇔ z∗ is a pole of H(z). (The same applies for zeros.) The connection to the transfer function: The pole-zero plot tells us where the transfer function is zero and infinite. In the regions surround- ing these points, the transfer function is low or high, respectively. Approximating the frequency response, determining the filter type: Use the angles of poles/zeroes to determine the angular frequency of peaks/dips in the spectrum, and use their distance from the unit circle to determine (approximately) the height of the peak or dip.

4

2.2 The inverse z-transform

Inverse z-transform of rational transfer functions by partial fraction expansion. Any transfer function in the form of a rational polynomial

H(z) = B(z)

A(z) = b0 + b1z

−1 + · · ·+ bMz−M

a0 + a1z−1 + · · ·+ aNz−N

can be reduced to the form

H(z) =

L∑ l=0

K−1∑ k=0

blz −l ck

1− pkz−1 .

and inverse transformed using the properties of linearity and time shifts.

3 Frequency analysis of signals

3.1 Continous-time signals

The (continous-time) Fourier series Synthesis equation: x(t) =

∑∞ k=−∞ cke

2πT0 kt =

∑∞ k=−∞ cke

2πF0kt.

Analysis equation: ck = 1Tp ∫ Tp x(t)e−

2π T0 ktdt = 1Tp

∫ Tp x(t)e−2πF0ktdt.

• All periodic signal of practical interest satisfy these conditions.

• Other periodic signals may also have a Fourier series representation.

The (Continous-Time) Fourier Transform (FT/CTFT) Synthesis equation: x(t) =

∫∞ −∞X(F )e

2πFtdF . Analysis equation: X(F ) =

∫∞ −∞ x(t)e

−2πFtdt.

• All signal of practical interest satisfy these conditions.

3.2 Discrete-time signals

The discrete-time Fourier series (almost DFT !!!) Synthesis equation: x[n] =

∑N−1 k=0 cke

2πkn/N . Analysis equation: ck = 1N

∑N−1 n=0 x[n]e

−2πkn/N .

• {ck} represents the amplitude and phase associated with the frequency component sk[n] = e

2πkn/N = ewkn, wk = 2πk/N .

• {ck} periodic with period N .

The discrete-time Fourier transform (DTFT) Synthesis equation: x[n] = 12π

∫ 2π X(eω)eωndω

Analysis equation: X(eω) = ∑∞ n=−∞ x[n]e

−ωn

• X(eω) unique over the frequency interval (−π, π), or equivalently, (0, 2π).

• X(eω) periodic with period 2π.

• Convergence: XN (eω) = ∑N n=−N x[n]e

−ωn converges uniformly to X(eω), i.e. limN→∞{supw |X(eω)−XN (eω)|} = 0.

– Guaranteed if x[n] is absolutely summable.

• Possible with square summable sequences if mean-square convergence condition.

5

Energy density spectrum Parseval relations:

Cont. time Disc.-time

Per Infinite energy and Px =

1 Tp

∫ Tp |x(t)|2 dt =

∑∞ k=−∞ |ck|2

Px = 1 N

∑N−1 n=0 |x[n]|2 =

∑N−1 k=0 |ck|2

Ex = ∑N−1 n=0 |x[n]|2 = N

∑N−1 k=0 |ck|2

Aper x(t) any finite enery signal with FT X(F ) Ex =

∫∞ t=−∞ |x(t)|

2 dt

= ∫∞ −∞ |X(F )|

2 dF

Ex = ∑∞ n=−∞ |x[n]|2

= 12∗π ∫ π −π |X(e

ω)|2 dω

The relationship between the Fourier transform and the z-transform X(eω) = X(z)|z=eω

The four Fourier series/transforms

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Properties of the Fourier transform

3.3 The frequency response function

The frequency response function

• H(eω) = H(z)|z=ew ∑∞ k=−∞ h[k]e

−kω .

• H(eω) is a function of the frequency variable w.

• H(eω) is, in general, complex-valued, and may be written as:

– Real and imaginary parts: H(eω) = HR(eω) + HI(eω) or – Magnitude and phase: H(eω) = |H(w)|eΘ(ω), – where |H(eω)|2 = H(eω)H∗(eω) = H2R(eω) +H2I (eω)

– and Θ(eω) = tan−1 HI(e ω)

HR(eω) .

• Group delay or envelope delay of H: τg(eω) = −dΘ(e ω)

dω .

• Periodicity: Since x[n] = enw0 = en(w0+2π), we must have that H(w0) = H(w0 + 2π).

• H(eω) exists if system is BIBO stable, i.e. ∑∞ n=−∞ |h[n]|

Computation of frequency response function

• H(eω) = b0 Π M k=1(1−zke

−w)

ΠNk=1(1−pke−w) = b0e

w(N−M) ΠMk=1(e w−zk)

ΠNk=1(e w−pk)

.

• If ew − zk = Vk(ω)eΘk(ω) ew − pk = Uk(ω)eΦk(ω)

• then |H(eω)| = |b0| V1(ω)···VM (ω)U1(ω)···UM (ω) ∠H(eω) = ∠b0 + w(N −M) + (Θ1(ω) + · · ·ΘM (eω))− (Φ1(ω) · · ·ΦN (ω)).

3.4 Ideal filters

Ideal filter characteristics

• Ideal filters have constant magnitude characteristic.

• Response characteristics of lowpass, highpass, bandpass, all-pass and bandstop or band- elimination filters.

• Linear phase response Ideal filters have linear phase in their passband.

• I

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