A system that is both linear and time-invariant is called linear time-invariant (LTI).

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1 The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Lecture Notes: Time, Frequency & Transform Domains February 28, 2012 Signals & Systems Signals are mapped to other signals by systems. The action of a system H with input signal x and output signal y is denoted as: y = Hx Here, signals and scalars are, in general, complex valued. Signals occupy a linear space, meaning if x 1, x 2 are signals and α 1, α 2 are scalars, then α 1 x 1 + α 2 x 2 is also a signal. Signals can be continuous-time, x (t) for t R, or discrete-time, x [n] for n Z. There are generalizations, such as multidimensional signals, that are not considered here. Definition 1 Linearity: We say H is linear if the following holds. For input signals x 1, x 2 and scalars α 1, α 2, if y 1 = Hx 1, y 2 = Hx 2 then: α 1 y 1 + α 2 y 2 = H (α 1 x 1 + α 2 x 2 ) Definition 2 Time-Invariance (TI): We say H is TI if the following holds. For arbitrary x, let y = Hx. In the continuous-time case, y (t t 0 ) = Hx (t t 0 ) for all t 0, and in the discrete-time case, y [n n 0 ] = Hx [n n 0 ] for all n 0. Equivalently, H commutes with a delay. A system that is both linear and time-invariant is called linear time-invariant (LTI). Definition 3 Causal System: We say a system H is causal if the following holds. Consider the continous-time case. For all t 0 : if x 1, x 2 are signals such that x 1 (t) = x 2 (t) for all t t 0, then y 1 = Hx 1, y 2 = Hx 2 satisfy y 1 (t) = y 2 (t) for all t t 0. Similarly in the discrete-time case. Notice that, in many cases, systems are by default causal. For example, a transistor amplifier circuit cannot be noncausal! There are cases where noncausal systems are studied (and, in the discrete-time case, sort of implemented). Definition 4 Causal Signal: A signal x is causal if it is zero for negative time: x (t) = 0 for t < 0, or x [n] = 0 for n 1, in the continuous-time and discrete-time cases, respectively. Note that a nonzero value at time 0 is permitted. Definition 5 Bounded signal: A signal x is bounded if there is a finite value M < such that x M for all time. Definition 6 BIBO Stability: A system H is bounded-input bounded-output (BIBO) stable if a bounded input always results in a bounded output. That is, if x M < for all time and y = Hx, then there exists M < such that y M < for all time. 1

2 Note that for a BIBO stable system, if y = Hx, the output does not have to have the same bound as the input, that is, in the above defnition, M > M is possible. Definition 7 Convolution: The convolution of two signals h, x is a signal y, denoted as y = h x, given by: continuous-time : y (t) = discrete-time : y [n] = τ= k= h (τ) x (t τ) dτ h [k] x [n k] In general, h x = x h and (g h) x = g (h x). The impulse is a signal, denoted as δ, such that: δ x = x δ = x In the discrete-time case, δ [n] = 1 for n = 0 and 0 for n 0. In the continous-time case, δ (t) is more diffi cult to describe, beyond the fact that it serves as the identity for convolution; δ (t) is unbounded. In each case, δ is a causal signal. We define the unit-step as u (t) = 1, t > 0, u (t) = 0, t < 0, for continuous-time (the value at t = 0 can be arbitrary); and for discrete-time, u [n] = 1, n 0 and u [n] = 0, n 1. The unit-step is convenient notationally. For example, if x (t) = e t for t > 0 and x (t) = 0 for t 0, we can write that as x (t) = e t u (t). Theorem 8 A system H is LTI iff there exists a signal h, called the impulse response of the system, such that y = Hx implies y = h x. Theorem 9 An LTI system H is a causal system iff its impulse response h is a causal signal. Definition 10 The L p -norm of a signal, for p = 1, 2,, denoted x p, is defined as follows: 1 continuous-time: x 1 = x (t) dt x 2 = x (t) 2 dt x = max x (t) discrete-time: x 1 = x [n] x 2 = x [n] 2 x = max x [n] 1 The L -norm actually has a more technical definition. For example, let x (t) = (1 exp ( t)) u (t). Then x = 1 even though its maximum does not exist (x (t) never actually equals 1 at any time). 2

3 The L p -norm can be defined for other values in p, but the cases p = 1, 2, are the ones most commonly used. Lemma 11 If y = h x then y h 1 x. Theorem 12 An LTI system H is BIBO stable iff its impulse resonse h has a finite L 1 - norm, i.e., h 1 <. Before continuing, we define absolute convergence of sums and integrals: Definition 13 The integral x (t) dt of x over a (finite or infinite) range A R is t A absolutely convergent if x (t) dt <. t A Definition 14 The sum n I x [n] of x over a (finite or infinite) set of indices I Z is absolutely convergent if n I x [n] <. Analytic Functions and Transfer Functions Below, we will describe transfer functions via Laplace transform for continuous-time case, and z-transform for discrete-time case. In each case, we obtain a function of a complex variable (denoted s for the Laplace transform, and z for the z-transform). A function of a complex variable that is differentiable is called analytic. Because a complex variable is, essentially, two real variables, being analytic is more restrictive than ordinary differentiability for a function of a single real variable. For example, the seemingly nice function f (z) = z (conjugation) is NOT ANALYTIC ANYWHERE!! There is NO signal x [n] whose z-transform will be z. Transfer functions are analytic, and conversely analytic functions are transfer functions. That is, an analytic function of a complex variable is the Laplace transform of a continuoustime signal (or z-transform or a discrete-time signal) In the remainder of this section, we consider analytic functions only. If X (z 0 ) = 0, we say z 0 is a zero. Unless X is constant 0, then z 0 must be a zero of finite multiplicity m, meaning X (z) = (z z 0 ) m g (z) where g (z) is analytic and g (z 0 ) 0. An isolated point where X (z) is not analytic is called a singularity. We will deal with only one type of singularity- a pole. If lim z z0 X (z) =, then z 0 is a pole. A pole must occur with finite multiplicity m, meaning X (z) = g (z) / (z z 0 ) m where g (z) is analytic and not 0 at z 0. Polynomials, trigonometric and exponential functions, and arithmetic compositions (e.g., division) of analytic functions are generally analytic except at the obvious places; for example, 1/ sin (z) has poles at z = nπ, n Z. When dealing with z-transform (NOT Laplace transform), we need to consider behavior at z =. Definition 15 X (z) is analytic at if X (z) is analytic in a region of the form z > R and lim z X (z) exists (is finite). As a special case, X (z) has a zero at if it is analytic at at the limit is 0. X (z) has a singularity at if it is analytic in a region of the form z > R but lim z X (z) does not exist. In particular, this singularity is a pole (i.e., X (z) has a pole at ) iff X is analytic in a region of the form z > R and lim z X (z) =. 3

4 Note that e z has a singularity at, but it is not a pole. Every analytic function must have a singularity somewhere (possibly at ), unless it is constant. Unless X is identically 0, a zero at must occur with finite multiplicity, m, meaning X (z) = g (z) /z m where g (z) is analytic but not 0 at. Also, a pole at must occur with finite multiplicity m, meaning X (z) = z m g (z) where g (z) is analytic but not 0 at. A rational function is a ratio of polynomials. Counting multiplicities, and counting poles or zeros at, the number of zeros equals the number of poles for a rational function. For example: has a double pole at. z z 2 + z + 1 Frequency and Transform Domains- Continuous-Time Case Here we assume all signals are continuous-time. Definition 16 Laplace transform: The Laplace transform of x (t) is denoted X (s) and is given by: X (s) = t= x (t) e st dt where, in general, s is complex. We write X = L {x} and x = L 1 {X}. We also write x X. The region of convergence (ROC) is the region in the complex s-plane where the integral converges absolutely. The general form of the ROC is a vertical strip R 1 < Re (s) < R 2 where R 1 = and/or R 2 = + is possible. Notice the boundaries are themselves not part of the ROC. In general, X (s) will have poles (or other singularities) on the borders. A single X (s) will be associated with different time domain signals x (t) in each ROC. That is: The inverse Laplace transform L 1 {X} is not uniquely determined unless we specify an associated ROC. Definition 17 Fourier transform: The continuous-time Fourier transform (CTFT) of x (t) is denoted X (ω) and is given by: X (ω) = x (t) e jωt dt The variable ω R, and is radian frequency, having units rad/sec. (ICTFT) is given by: x (t) = 1 X (ω) e jωt dt 2π The inverse CTFT 4

5 We can relate ω to frequency f in Hertz via ω = 2πf, and can rewrite the CTFT and ICTFT as: X (f) = x (t) = x (t) e j2πft dt X (f) e j2πft dt In the remainder of these notes, we work with radian frequency, ω, not f. We write X = F {x} and x = F 1 {X}. We also write x X. Sometimes to distinguish the Laplace and Fourier transforms, we may write X CT F T (ω) for the CTFT. The following, however, suggests a more common convention. Theorem 18 The Laplace transform X (s) and CTFT X CT F T (ω) are related via the algebraic substitution s = jω, that is: X CT F T (ω) = X (jω) Lemma 19 If y = h x then Y (s) = H (s) X (s) and Y (jω) = H (jω) X (jω). Theorem 20 For each fixed ω R, the complex sinewave φ ω (t) = e jωt is an eigenfunction for every stable LTI system H, that is: Hφ ω = λφ ω where λ is a complex scalar, that (in general) depends on ω. In particular, λ = H (jω), the CTFT of the impulse response h evaluated at ω. We call H (jω) the frequency response of the system, and the main result is that the frequency response is the Fourier transform of the impulse response. If the system is unstable, a sinusoidal input could produce an unbounded output, so the frequency response is not well-defined. Theorem 21 If H is an LTI system, then y = Hx implies Y (jω) = H (jω) X (jω), that is, the Fourier transform of the output is the Fourier transform of the input multiplied by the frequency response of the system. Theorem 22 If H is an LTI system, then y = Hx implies Y (s) = H (s) X (s), that is, the Laplace transform of the output is the product of the Laplace transform of the input and the function H (s), called the transfer function of the system. The transfer function is the Laplace transform of the impulse response. Note that the system is governed by multiplication in both the frequency and transform domains. Normally, we call the Fourier transform of a signal X (jω) a spectrum, and the Fourier transform of the impulse response of a system H (jω) a frequency response. However, we will often use the term "frequency response" to apply to any Fourier transform, even if it represents a signal (we could, for example, consider the system whose impulse response is the prescribed signal this principle is called the duality of signals and systems). If H (jω) = H (jω) exp (jφ (ω)), then H (ω) is called the magnitude response and φ (ω) is called the phase response. We usually deal with signals and sytems whose Laplace transform is a ratio of polynomials. 5

6 Definition 23 A function H (s) that is a ratio of polynomials is called a rational function. Specifically, if H (s) = B (s) /A (s), the roots of the numerator polynomial B (s) are called the zeros of the system, and the roots of the denominator polynomials A (s) are called the poles of the system. From now on, we assume all signals and systems have Laplace transforms that are rational functions with real coeffi cients. This means that poles and zeros, if they are complex, occur in complex conjugate pairs. Notice that the imaginary axis in the s-plane can be viewed as the frequency domain, i.e., the substitution s = jω locates the frequency domain in the s-plane. The imaginary axis divides the s-plane into two regions: the left-hand plane (LHP), where Re (s) < 0, and the right-hand plane (RHP), where Re (s) > 0. Theorem 24 (Properties of ROC and Pole Locations for Laplace Transform) If σ 1 < σ 2 < < σ M are the distinct values of the real parts of the poles, then the ROCs for H (s) are the vertical strips Re (s) < σ 1, and σ i < Re (s) < σ i+1 for 1 i M 1, and Re (s) > σ M. For each ROC, there is a distinct time-domain signal h (t) whose Laplace transform is H (s). Two special cases of interest: a causal signal (or system) corresponds to the rightmost ROC, Re (s) > σ M ; the ROC that include the jω-axis (i.e., Re (s) = 0) corresponds to signals with well-defined CTFT (i.e., the Fourier transform integral is absolutely convergent), and, for a system, represent stable systems and those with well-defined frequency response. In other words, the substitution s = jω that relates the Laplace and Fourier transforms is permitted only in the particular ROC in the s-plane that includes the jω-axis. In particular, if H (s) has a pole on the jω-axis, then it cannot represent a stable system (the purported frequency response would blow up at the frequency where the pole is located). Theorem 25 A causal system is stable iff all its poles are in the LHP. Theorem 26 If H (s) is rational with distinct poles p 1, p 2,, p M with multiplicities m 1, m 2,, m M, then the general form of the causal impulse response h (t) is: h (t) = K (t) + M c i (t) exp (p i t) u (t) i=1 where K (t), if present, contains impulsive signals (like δ (t)), called the singular part, and c i (t) is a polynomial in time of degree m i 1. Based on the above, Re (p i ) determines the time constant of exponential growth or decay, and Im (p i ) determines the frequency of oscillation (if p i is real, there is no oscillation), for each term in the sum, called a mode. In general, K (t) will be present only if the numerator of H (s) has degree greater than or equal to that of the denominator. Simple poles (multiplicity 1) at σ ± jω yields exp ((σ ± jω) t) terms, which can be combined as e σt cos (ωt + θ). Comparing e σt with e ±t/τ, we see the following: If σ = Re (p) > 0, we have exponential growth. If σ = Re (p) < 0 we have exponential decay. In either case, the time constant is τ = 1/ σ (seconds) 6

7 If ω = Im (p) 0, the exponential growth or decay is accompanied by oscillation at a radian frequency ω (rad/sec). Purely real poles yield no oscillation. As a special case, if there is a pole at s = 0, then h (t) contains a term of the form a m 1 t m a 1 t + a 0, t 0, where m is the multiplicity of this pole. If there are poles at ±jω 0, then h (t) contains a term of the form cos (ω 0 t + θ), if the poles are simple, or a polynomial times a sinewave if the poles have multiplicity more than 1. Definition 27 The partial fraction expansion of rational H (s), with distinct poles p i with multiplicities m i (1 i M) is the expression: H (s) = K (s) + M i=1 m i 1 k=0 r ik (s p i ) k where K (s) is a polynomial in s (present only if the numerator of H has a degree greater or equal to that of the degree of the denominator). The coeffi cients r ik are called the residues. The K (s) term, if it is present, contributes to the singular part of h (t). The terms in the partial fraction expansion { } for a particular pole p i contribute to the respective mode in h (t). For example, L 1 1 = e pt u (t). s p We conclude with some basic properties of Laplace and Fourier transforms: Theorem 28 Derivative: x (t) sx (s), more generally x (n) (t) s n X (s); similarly, x (t) jωx (jω), and x (n) (t) (jω) n X (jω) Continuous-time systems that give rise to rational transfer functions H (s) can be represented with a finite number of differentiators (or integrators), combined with constant multipliers and summers; these give rise to ordinary differential equations. These are called lumped systems. Systems that are not characterized by rational transfer functions are distributed systems, and are generally governed by partial differential equations (such as the wave equation). For example, a delay is a distributed system (in continuous-time), as the following theorem shows; in practice, high frequency circuits (say in the microwave or optical range) can realize delays, but ordinary circuits comprised say of resistors, inductors, capacitors and/or op-amps cannot achieve a pure delay. Theorem 29 Delay: h (t t 0 ) e jωt 0 H (jω) Observe the corresponding Laplace transform relationship is h (t t 0 ) e st 0 H (s), which is not rational. Note that a delay does not change the magnitude response, but adds a phase response, called a linear phase factor, which is a linear function of frequency. Specifically, the slope of the phase response is negative, and equals the delay. 7

8 Frequency and Transform Domains- Discrete-Time Case Here we assume all signals are discrete-time. Definition 30 z-transform: The z-transform of x [n] is denoted X (z) and is given by: X (z) = n= x [n] z n where, in general, z is complex. We write X = Z {x} and x = Z 1 {X}. We also write x X. The region of convergence (ROC) is the region in the complex z-plane where the sum converges absolutely. The general form of the ROC is R 1 < z < R 2 where R 1 = 0 and/or R 2 = + is possible. A region of this form is called an annular ring. Special cases are 0 z < R (if X (z) is analytic at z = 0) and R < z (if X (z) is analytic at ). For example, if X is NOT analytic at, then the outermost ROC has the form R < z <. The boundary circle is never part of the ROC itself; X (z) will have poles (or other singularities) on the boundaries. As with Laplace transforms, the z-transform is analytic in the ROC, and X (z) represents the z-transform of some time-domain signal x [n] only if it is analytic in an annular ring. Also, each ROC corresponds to a different time-domain signal x [n]. That is: The inverse z-transform Z 1 {X} is not uniquely determined unless we specify an associated ROC. When dealing with z-transforms, it is important to consider the behavior of X (z) at z =. That is, we must specify whether the X (z) is analytic at, if there is a zero at (with particular multiplicity), or if is a pole (again, with prescribed multiplicity). WE DO NOT EXAMINE WHETHER THERE ARE POLES OR ZEROS AT IN THE CASE OF LAPLACE TRANSFORM. Definition 31 Fourier transform: The discrete-time Fourier transform (DTFT) of x [n] is denoted X (ω) and is given by: X (ω) = x [n] e jωn The variable ω R, and is normalized digital radian frequency, having units rad (NOT rad/sec!!). The inverse DTFT (IDTFT) is given by: x [n] = 1 2π π π X (ω) e jωn dω The DTFT is always PERIODIC in ω with period 2π. 8

9 Note that X (ω) must be periodic with period 2π. If it fails to be periodic, it cannot represent the DTFT of any discrete-time signal! The bounds of the IDTFT integral can, in fact, be any interval of length 2π. Recall that exp (jθ) = 1 iff θ = 2πn, n Z. Thus, if we define the complex sinewave at frequency ω as the signal φ ω [n] = exp (jωn), observe that φ ω = φ ω+2π. Thus, we need only consider ω over an interval of length 2π (extended then periodically to all of R), for example, ω [ π, π), i.e., { π ω < π}. We write X = F {x} and x = F 1 {X}, or x X. Sometimes to distinguish the z- and Fourier transforms, we may write X DT F T (ω) for the DTFT. The following, however, suggests a more common convention. Theorem 32 The z-transform X (z) and DTFT X DT F T (ω) are related via the algebraic substitution z = e jω, that is: X DT F T (ω) = X ( e jω) Lemma 33 If y = h x then Y (z) = H (z) X (z) and Y (e jω ) = H (e jω ) X (e jω ). Theorem 34 For each fixed ω R, the complex sinewave φ ω [n] = e jωn is an eigenfunction for every stable LTI system H, that is: Hφ ω = λφ ω where λ is a complex scalar, that (in general) depends on ω. In particular, λ = H (e jω ), the DTFT of the impulse response h evaluated at ω. We call H (e jω ) the frequency response of the system, and the main result is that the frequency response is the Fourier transform of the impulse response. If the system is unstable, a sinusoidal input could produce an unbounded output, so the frequency response is not well-defined. Theorem 35 If H is an LTI system, then y = Hx implies Y (e jω ) = H (e jω ) X (e jω ), that is, the Fourier transform of the output is the Fourier transform of the input multiplied by the frequency response of the system. Theorem 36 If H is an LTI system, then y = Hx implies Y (z) = H (z) X (z), that is, the z-transform of the output is the product of the z-transform of the input with the function H (z), called the transfer function of the system. The transfer function is the z-transform of the impulse response. Note that the system is governed by multiplication in both the frequency and transform domains. We usually deal with signals and sytems whose z-transforms are rational functions. From now on, we assume all signals and systems have z-transforms that are rational functions with real coeffi cients. This means that poles and zeros, if they are complex, occur in complex conjugate pairs. Notice that the unit circle ( z = 1) in the z-plane can be viewed as the frequency domain, i.e., the substitution z = e jω locates the frequency domain in the z-plane, and in fact the geometry of the unit circle is consistent with the 2π-periodicity of the frequency domain. The unit circle divides the z-plane into two regions: { z < 1} and { z > 1}. 9

10 Theorem 37 (Properties of ROC and Pole Locations for Laplace Transform) If R 1 < R 2 < < R M are the distinct values of the magnitudes of the poles, then the ROCs for H (z) are z < R 1 (or 0 < z < R 1 if there is a pole at z = 0), and R i < z < R i+1 for 1 i M 1, and R M < z <, or R M < z if X (z) is analytic at. For each ROC, there is a distinct time-domain signal h [n] whose z-transform is H (z). That is, Z 1 {H} is not uniquely defined unless we specify an ROC! Two special cases of interest: a causal signal (or system) corresonds to the ROC R M < z (specifically, if H (z) has a causal inverse transform only if it is analytic at and, in this case, the corresponding ROC is the outermost); and the ROC that includes the unit circle ( z = 1) corresponds to signals with well-defined DTFT (i.e., the Fourier transform sum is absolutely convergent), and, for a system, represents a stable system and (equivalently) one with a well-defined frequency response. In other words, the substitution z = e jω that relates the z- and Fourier transforms is permitted only in the particular ROC in the z-plane that includes the unit circle. In particular, if H (z) has a pole on the unit circle, then it cannot represent a stable system (the purported frequency response would blow up at the frequency where the pole is located). A basic operation in discrete-time systems is a delay. Here is the fundamental result: Theorem 38 Transform of a delay: h [n 1] z 1 H (z), and, more generally, h [n n 0 ] z n 0 H (z). Also, h [n n 0 ] e jωn 0 H (e jω ) Digital filters are comprised of a fixed number of delays (representing, in fact, memory), combined with multiplication with fixed coeffi cients and summers. All such systems give rise to causal rational transfer functions H (z). Note that a delay does not change the magnitude response of a system, but adds a linear-phase factor (phase response a linear function of frequency), where the slope of the phase response is the amount of delay. Now, if H (z) is analytic at, then the outermost region, R < z, corresponds to causal h [n]. We can see this, for example, from the z-transform sum itself: h [n] for n 1 corresponds to positive powers of z, which blow up at. If h [n] = 0 for all n 1, then H (z) h [0] as z. Notice h [0] 0 then means H (z) is analytic at but does not have a zero there. What if H (z) has a pole of multiplicity m at? Then z m H (z) is analytic but not 0 at ; hence h [n m] = 0 for n 1 but h [n m] 0 for n = 0. Therefore, the support (range where h is nonzero) is n m; this is called a right-sided signal, and can be made causal by a finite delay. Theorem 39 A causal system is stable iff all its poles are inside the unit circle ( z < 1). Theorem 40 If H (z) is rational with distinct finite poles p 1, p 2,, p M with multiplicities m 1, m 2,, m M, then the general form of the right-sided impulse response h [n] is: h [n] = K [n] + M c i [n] p n i u [n] i=1 where K [n] has the form: K [n] = k 0 δ [n] + k 1 δ [n + 1] + k N δ [n + N] 10

11 if H (z) has a pole of multiplicity N at (otherwise, if H (z) is analytic at and thus h [n] is causal, K [n] = k 0 δ [n]), and c i [n] is a polynomial in time of degree m i 1. Based on the above, p i determines the time constant of exponential growth or decay, and arg (p i ) determines the frequency of oscillation (if p i is positive real, there is no oscillation), for each term in the sum, called a mode. For example, simple poles at p = αe ±jω (multiplicity 1) yields α n e ±jωn terms, which can be combined as α n cos (ωt + θ). Comparing α n with e ±t/τ, assuming t = nt (T is the sample period) we see the following: If α = p > 1, we have exponential growth. If α = p < 1 we have exponential decay. In either case, the time constant is τ = T/ ln p (seconds). If ω = arg (p) 0, the exponential growth or decay is accompanied by oscillation at a radian frequency ω (rad/sec). In particular, if p is negative real, p = p e jπ and there is oscillation at frequency π (e.g., ( 1/2) n has alternating signs). As a special case, if there is a pole at z = 1 = e j0, then h [n] contains a term of the form (a m 1 n m a 0 ) u [n], where m is the multiplicity of this pole. If there are poles at e ±jω 0, then h [n] contains a term of the form cos (ω 0 n + θ), if the poles are simple, or a polynomial times a sinewave if the poles have multiplicity more than 1. Definition 41 The partial fraction expansion of rational H (z), with distinct poles p i with multiplicities m i (1 i M) is the expression: H (z) = K (z) + M i=1 m i 1 k=0 r ik z (z p i ) k where K (z) is a polynomial in z, and the coeffi cients r ik are called the residues. In general, K (z) is always at least a constant term. If the numerator has degree STRICTLY greater than that of the denominator, then K (z) has degree equal to the number of poles of H (z) at. The terms in the partial fraction expansion { } for a particular pole p i contribute to the respective mode in h [n]. For example, Z 1 z = p n u [n]. Frequency Response Graphs z p Lemma 42 x (t) X ( ω), and x ( t) X (ω), in the continuous-time case, and x [n] X ( ω), and x [ n] X (ω) in the discrete-time case. Definition 43 A function f (ξ) is conjuugate symmetric if f (ξ) = f ( ξ). Equivalent conditions for conjugate symmetry: the real part and magnitude are even functions (e.g., f (ξ) = f ( ξ) ), and the imaginary part and phase are odd functions (e.g., arg f (ξ) = arg f ( ξ)). Note that f (ξ) is real iff f (ξ) = f (ξ). 11

12 Theorem 44 A signal is real in the time-domain iff it is conjugate symmetric in the frequency domain. Theorem 45 A signal has a real spectrum (equivalently, a system has zero-phase response) iff it is conjugate symmetric in the time domain. More generally, a signal has a linear phase response iff it exhibits conjugate symmetry around some fixed time x (t) = x (T t 0 ), in which case the phase response φ (ω) = φ (0) (T/2) ω, or x [n] = x [N n], in which case the phase response φ (ω) = φ (0) (N/2) ω. Since we usually deal with signals and systems that are real, we normally graph the frequency response for positive frequencies only (0 to in the analog case, and 0 to π in the digital case); if a real signal has a component at a positive frequency, it also has a component at the negative frequency, so we normally don t report or list the negative frequencies explicitly. For example, a signal with bandwidth B generally means X (ω) = 0 for ω > B, i.e., the signal occupies the range B < ω < B, which actually has width 2B! The magnitude response is often graphed on a decibel scale, specifically 20 log 10 H (using 20, not 10, because H represents a ratio of signal amplitudes, not power). But magnitude on a linear scale is used as well. Phase is usually graphed in degrees, unwrapped so as to be a continuous curve. If φ (ω) is computed without unwrapping, its graph may contain jumps of 2π (radians), or multiples of it. Unwrapping removes these. HOWEVER, one may still observe jump discontinuities in the graph of φ (ω), even after unwrapping. These are jump discontinuities of π (i.e., 180 ). Why do they happen? If H (ω 0 ) = 0 at some frequency, then the phase response φ (ω 0 ) = arg H (ω 0 ) is technically undefined. Consider for example, H (ω) going through a zero-crossing at ω 0, say changing from to Then the phase jumps from 0 to 180. So these zeros in the frequency response lead to 180 jump discontinuities in the phase response, that cannot be removed. Normally, the slope of the phase response represents a delay, but these jump discontinuities are not problematic because the system is zeroing out signals at or near these frequencies, anyway. It should be noted this is a common occurrence in digital filters, and occurs much less often with analog systems; however, it can indeed occur in the analog case, and is not a phenomenon unique to digital filters. 12

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