The Complex-Baseband Equivalent Channel Model

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ECE-501 Phil Schniter February 4, 2008 replacements Complex-Baseband Equivalent Channel: Linear communication schemes (e.g., AM, QAM, VSB) can all be represented (using complex-baseband mod/demod) as:

wideband modulation demodulation channel s(t) r(t) m˜ (t) Re h(t) + LPF v˜(t) × ×

j2πfct w(t) j2πfct ∼ e ∼ 2e− It turns out that this diagram can be greatly simpliﬁed. . .

First, consider the signal path on its own:

s(t) x(t) m˜ (t) Re h(t) LPF v˜s(t) × ×

j2πfct j2πfct ∼ e ∼ 2e− Since s(t) is a bandpass signal, we can replace the wideband

channel response h(t) with its bandpass equivalent hbp(t):

S(f) S(f) | | | | 2W f f

f f

33 ECE-501 Phil Schniter February 4, 2008

Then, notice that

j2πfct j2πfct s(t) h (t) 2e− = s(τ)h (t τ)dτ 2e− ∗ bp bp − · # ! " j2πfcτ j2πfc(t τ) = s(τ)2e− h (t τ)e− − dτ bp − # j2πfct j2πfct = s(t)2e− h (t)e− , ∗ bp which means we can rewrit!e the block di"agr!am as "

s(t) j2πfct m˜ (t) Re hbp(t)e− LPF v˜s(t) × ×

j2πfct j2πfct ∼ e ∼ 2e−

j2πfct We can now reverse the order of the LPF and hbp(t)e− (since both are LTI systems), giving

s(t) j2πfct m˜ (t) Re LPF hbp(t)e− v˜s(t) × ×

j2πfct j2πfct ∼ e ∼ 2e− Since mod/demod is transparent (with synched oscillators), it can be removed, simplifying the block diagram to

j2πfct m˜ (t) hbp(t)e− v˜s(t)

34 ECE-501 Phil Schniter February 4, 2008

Now, since m˜ (t) is bandlimited to W Hz, there is no need to j2πfct model the left component of Hbp(f + f ) = hbp(t)e− : c F{ }

f f f fc fc 2fc 0 B 0 B − − − j2πfct Replacing hbp(t)e− with the complex-baseband response h˜(t) gives the “complex-baseband equivalent” signal path:

m˜ (t) h˜(t) v˜s(t)

The spectrums above show that h (t) = Re h˜(t) 2ej2πfct . bp { · } Next consider the noise path on it’s own:

LPF(f) w(t) LPF v˜ (t) | | × n

j2πfct f 2e Bs Bp W W Bp Bs ∼ − − − − From the diagram, vñ(t) is a W (f) baseband version of the band- | | f f f pass noise spectrum that oc- − c c W (f + f ) cupies the frequency range | c | f [f B , f + B ]. f c s c s B B ∈ − − s s Vñ(f) Since vñ(t) is complex-valued, | | ˜ f Vn(f) is non-symmetric. B B − s s

35 ECE-501 Phil Schniter February 4, 2008

Say that w(t) is real-valued white noise with power spectral density (PSD) Sw(f) = N0. Since Sw(f) is constant over all f, the PSD of the complex noise v˜n(t) will be constant over the LPF passband, i.e., f [ B , B ]: ∈ − p p

Sv˜n (f)

f B B W W B B − s − p − p s A well-designed communications receiver will suppress all energy outside the signal bandwidth W , since it is purely noise. Given that the noise spectrum outside f [ W, W ] ∈ − will get totally suppressed, it doesn’t matter how we model it!

Thus, we choose to replace the lowpass complex noise v˜n(t) with something simpler to describe: white complex noise w˜(t) with PSD Sw˜(f) = N0:

Sw˜(f)

f B B W W B B − s − p − p s We’ll refer to w˜(t) as “complex baseband equivalent” noise.

36 ECE-501 Phil Schniter February 4, 2008

Putting the signal and noise paths together, we arrive at the complex baseband equivalent channel model:

complex baseband equivalent channel ˜ j2πfct hbp(t) = Re h(t) 2e m˜ (t) h˜(t) + v˜(t) { · } w˜(t) = complex-valued white noise w˜(t)

complex baseband bandpass equiv complex baseband modulation channel demodulation ⇔ s(t) r(t) m˜ (t) Re hbp(t) + LPF v˜(t) × ×

j2πfct w(t) j2πfct ∼ e ∼ 2e−

I/Q modulation wideband I/Q demodulation ⇔ channel m (t) LPF v (t) I × × I cos(2πfct) s(t) r(t) cos(2πfct) + h(t) + ∼ ∼ sin(2πfct) sin(2πfct) − m (t) w(t) − LPF v (t) Q × × Q

The diagrams above should convince you of the utility of the complex-baseband representation in simplifying the system model!