Discrete-Time Fourier Transform

BIBLIOGRAPHY

2.4 Discrete-Time Fourier Transform

for causality is thath[n]=0, forn <0. One can argue this necessary and sufficient condition by exploring the signal-flip interpretation of convolution (Exercise 2.2). A consequence of causality is that ifx₁[n]=x₂[n] forn < n_o, theny₁[n]=y₂[n] forn < n_o.

EXAMPLE2.1 The two properties of stability and causality are illustrated with an LTI system having an exponentially decaying impulse responseh[n]=Aaⁿforn≥0 and zero otherwise. The system is causal becauseh[n]=0 forn < 0 and, when|a|<1, the system is stable because the impulse response is absolutely summable:

∞ n=−∞

|h[n]| = A ∞ n=0

|a|ⁿ

= A 1− |a| where we have used the geometric series relation_∞

n=0bⁿ= _1−b¹ for|b|<1. If, on the other hand,

|a| ≥1, then the geometric series does not converge, the response is not absolutely summable, and the system is unstable. Note that, according to this condition, a system whose impulse response is the

unit step function, i.e.,h[n]=u[n], is unstable. 䉱

The terminology formulated in this section for systems is also used for sequences, although its physical meaning for sequences is lacking. Astable sequenceis defined as an absolutely summable sequence, and acausal sequenceis zero forn <0. A causal sequence will also be referred to as aright-sided sequence.

2.4 Discrete-Time Fourier Transform

The previous section focused on time-domain representations of signals and systems. Frequency- domain representations, the topic of this section, are useful for the analysis of signals and the design of systems for processing signals. We begin with a review of the Fourier transform.

A large class of sequences can be represented as a linear combination of complex expo- nentials whose frequencies lie in the range²[−π, π]. Specifically, we write the following pair of equations:

x[n] = ¹ 2π

_π

−πX(ω)e^{j ωn}dω X(ω) =

∞

n=−∞

x[n]e^{−j ωn}. (2.2)

This pair of equations is known as thediscrete-time Fourier transformpair representation of a sequence. For convenience, the phraseFourier transformwill often be used in place ofdiscrete-

2Recall thate^{j (ω+2π )n}=e^{j ωn}.

time Fourier transform. Equation (2.2) represents x[n] as a superposition of infinitesimally small complex exponentialsdωX(ω)e^{j ωn}, whereX(ω)determines the relative weight of each exponential.X(ω)is the Fourier transform of the sequencex[n], and is also referred to as the

“analysis equation” because it analyzesx[n] to determine its relative weights. The first equation in the pair is theinverse Fourier transform, also referred to as the “synthesis equation” because it puts the signal back together again from its (complex exponential) components. In this text, we often use the terminology “analysis” and “synthesis” of a signal. We have not yet explicitly shown for what class of sequences such a Fourier transform pair exists. Existence means that (1) X(ω)does not diverge, i.e., the Fourier transform sum converges, and (2)x[n] can be obtained fromX(ω). It can be shown that a sufficient condition for the existence of the pair is thatx[n] be absolutely summable, i.e., thatx[n] is stable [7]. Therefore, all stable sequences and stable system impulse responses have Fourier transforms.

Some useful properties of the Fourier transform are as follows (Exercise 2.3):

P1:Since the Fourier transform is complex, it can be written in polar form as X(ω) = X_r(ω)+ j X_i(ω)

= |X(ω)|e^{j X(ω)}

where the subscriptsrandidenote real and imaginary parts, respectively.

P2:The Fourier transform is periodic with period 2π: X(ω+2π ) = X(ω)

which is consistent with the statement that the frequency range [−π, π] is sufficient for repre- senting a discrete-time signal.

P3:For a real-valued sequencex_[n], the Fourier transform is conjugate-symmetric:

X(ω) = X^∗(−ω)

where∗denotes complex conjugate. Conjugate symmetry implies that the magnitude and real part ofX(ω)are even, i.e.,|X(ω)| = |X(−ω)|andX_r(ω)=X_r(−ω), while its phase and imaginary parts are odd, i.e., X(ω)= − X(−ω)andX_i(ω)= −X_i(−ω). It follows that if a sequence is not conjugate-symmetric, then it must be a complex-valued sequence.

P4:The energy of a signal can be expressed by Parseval’s Theorem as

∞

n=−∞

|x[n]|² = ¹ 2π

π

−π

|X(ω)|²dω (2.3)

which states that the total energy of a signal can be given in either the time or frequency domain.

The functions|x[n]|^{2 and}|X(ω)|²are thought of asenergy densities, i.e., the energy per unit time and the energy per unit frequency, because they describe the distribution of energy in time and frequency, respectively. Energy density is also referred to aspowerat a particular time or frequency.

2.4 Discrete-Time Fourier Transform 17

EXAMPLE2.2 Consider the shifted unit sample x[n] = δ[n−no]. The Fourier transform ofx_[n] is given by

X(ω) = ∞ n=−∞

δ[n−no]e^{−j ωn}

= e^{−j ωno}

sincex[n] is nonzero for only n= no. This complex function has unity magnitude and a linear phase of slope−no. In time, the energy in this sequence is unity and concentrated atn=no, but in frequency the energy is uniformly distributed over the interval [−π, π] and, as seen from Parseval’s

Theorem, averages to unity. 䉱

More generally, it can be shown that the Fourier transform of a displaced sequencex[n−n_o] is given byX(ω)e^{−j ωno}. Likewise, it can be shown, consistent with the similar forms of the Fourier transform and its inverse, that the Fourier transform ofe^{j ωon}x[n] is given byX(ω−ω_o). This later property is exploited in the following example:

EXAMPLE2.3 Consider the decaying exponential sequence multiplied by the unit step:

x[n] = aⁿu[n]

withagenerally complex. Then the Fourier transform ofx[n] is given by

X(ω) = ∞

n=0

aⁿe^{−j ωn}

= ∞

n=0

(ae^{−j ω})ⁿ

= ¹

1−ae^{−j ω}, |ae^{j ω}| = |a| < 1

so that the convergence condition onabecomes|a|<1. If we multiply the sequence by the complex exponentiale^{j ωon}, then we have the following Fourier transform pair:

e^{j ωon}aⁿu[n] ↔ ¹

1−ae^{−j (ω−ωo)}, |a| < 1.

An example of this later transform pair is shown in Figure 2.2a,b where it is seen that in frequency the energy is concentrated aroundω=ω_o= ^π₂. The two different values ofashow a broadening of the Fourier transform magnitude with decreasingacorresponding to a faster decay of the exponential.

From the linearity of the Fourier transform, and using the above relation, we can write the Fourier transform pair for a real decaying sinewave as

2aⁿcos(ωon)u[n] ↔ ¹

1−ae^{−j (ω−ωo)} + ¹

1−ae^{−j (ω+ωo)}, |a| < 1

6 5 4 3

AmplitudePhase (Radians) Phase (Radians)

2 1 0

2 1 0 –1 –2

6 5 4 3

Amplitude 2

1 0

2 1 0 –1 2 –2

0 –2

2 0

(a)

Radian Frequency Radian Frequency

(b) –2

2 0

–2

2 0

(c)

(d) –2

Figure 2.2 Frequency response of decaying complex and real exponentials of Example 2.3: (a) magnitude and (b) phase for complex exponential; (c) magnitude and (d) phase for decaying sinewave (solid for slow decay [a=0.9] and dashed for fast decay [a=0.7]). Frequencyωo=π/2.

where we have used the identity cos(α)= ¹₂(e^{j α}+e^{−j α}). Figure 2.2c,d illustrates the implications of conjugate symmetry on the Fourier transform magnitude and phase of this real sequence, i.e., the magnitude function is even, while the phase function is odd. In this case, decreasing the value ofa broadens the positive and negative frequency components of the signal around the frequenciesω_oand

−ωo, respectively. 䉱

Example 2.3 illustrates a fundamental property of the Fourier transform pair representation: A signal cannot be arbitrarily narrow in time and in frequency. We return to this property in the following section.

The next example derives the Fourier transform of the complex exponential, requiring in frequency the unit impulse which is also called the Dirac delta function.

2.4 Discrete-Time Fourier Transform 19

EXAMPLE2.4 In this case we begin in the frequency domain and perform the inverse Fourier transform. Consider a train of scaled unit impulses in frequency:

X(ω) = ∞ r=−∞

A2π δ(ω−ωo+r2π )

where 2π periodicity is enforced by adding delta function replicas at multiples of 2π (Figure 2.3a).

The inverse Fourier transform is given by³ x[n] = ¹

2π π

−π

A2π δ(ω−ωo)e^{j ωn}dω

= Ae^{j ωon}

which is our familiar complex exponential. Observe that this Fourier transform pair represents the time-frequency dual of the shifted unit sampleδ[n−no] and its transforme^{j noω}. More generally, a shifted Fourier transformX(ω−ωo)corresponds to the sequencex[n]e^{j ωon}, a property alluded to

earlier. 䉱

ω0 –ω0

–ω0

–ω1

–ω2

ω0

ω0ω1ω2

ω0ω1ω2ω3

a₀ a₀

a₀ a₂ a₂ a₂

a₁ a₁

a₁ a₃

ω ω

ω π

–π –π

–π –π

π

π ω

π X(ω)

X(ω) (a)

(c)

(b)

(d)

A A

X(ω)

X(ω)

Figure 2.3 Dirac delta Fourier transforms of (a) complex exponential sequence, (b) sinusoidal sequence, (c) sum of complex exponentials, and (d) sum of sinusoids. For simplicity,π and 2π factors are not shown in the amplitudes.

3Although this sequence is not absolutely summable, use of the Fourier transform pair can rigorously be justified using the theory of generalized functions [7].

Using the linearity of the Fourier transform, we can generalize the previous result to a sinusoidal sequence as well as to multiple complex exponentials and sines. Figure 2.3b–d illustrates the Fourier transforms of the following three classes of sequences:

Sinusoidal sequence

Acos(ω_on+φ) ↔ π Ae^{j φ}δ(ω − ω_o)+ π Ae^{−j φ}δ(ω +ω_o) Multiple complex exponentials

N

k=⁰

a_ke^{j ωkn+φk} ↔

N

k=⁰

2π a_ke^{j φk}δ(ω −ω_k) Multiple sinusoidals

N

k=0

a_kcos(ω_kn+ φ_k) ↔

N

k=0

π a_ke^{j φk}δ(ω −ω_k)+π a_ke^{−j φk}δ(ω +ω_k) For simplicity, each transform is represented over only one period; for generality, phase offsets are included.