14: FM Radio Receiver

14: FM Radio Receiver


• FM Radio Block Diagram

• Aliased ADC

• Channel Selection

• Channel Selection (1)



• FM Demodulator

• Differentiation Filter

• Pilot tone extraction +

• Polyphase Pilot tone

• Summary

DSP and Digital Filters (2017-10178) FM Radio: 14 – 1 / 12

FM Radio Block Diagram



• Aliased ADC





• FM Demodulator




• Summary


FM spectrum: 87.5 to 108MHz

[This example is taken from Ch 13 of Harris: Multirate Signal Processing]




• Aliased ADC





• FM Demodulator




• Summary


FM spectrum: 87.5 to 108MHzEach channel: ±100 kHz





• Aliased ADC





• FM Demodulator




• Summary



Baseband signal:Mono (L + R): ±15 kHz





• Aliased ADC





• FM Demodulator




• Summary



Baseband signal:Mono (L + R): ±15 kHz

Stereo (L – R): 38± 15 kHz





• Aliased ADC





• FM Demodulator




• Summary



Baseband signal:Mono (L + R): ±15 kHzPilot tone: 19 kHzStereo (L – R): 38± 15 kHz





• Aliased ADC





• FM Demodulator




• Summary



Baseband signal:Mono (L + R): ±15 kHzPilot tone: 19 kHzStereo (L – R): 38± 15 kHzRDS: 57± 2 kHz





• Aliased ADC





• FM Demodulator




• Summary




FM Modulation:





• Aliased ADC





• FM Demodulator




• Summary




FM Modulation:Freq deviation: ±75 kHz





• Aliased ADC





• FM Demodulator




• Summary




FM Modulation:Freq deviation: ±75 kHz

L–R signal is multiplied by 38 kHz to shift it to baseband


Aliased ADC



• Aliased ADC





• FM Demodulator




• Summary


FM band: 87.5 to 108MHzNormally sample at fs > 2f

Aliased ADC



• Aliased ADC





• FM Demodulator




• Summary



However:

Aliased ADC



• Aliased ADC





• FM Demodulator




• Summary



However:fs = 80MHz aliases banddown to [7.5, 28]MHz.

Aliased ADC



• Aliased ADC





• FM Demodulator




• Summary




–ve frequencies aliasto [−28, −7.5]MHz.

Aliased ADC



• Aliased ADC





• FM Demodulator




• Summary





We must suppress otherfrequencies that alias to therange ±[7.5, 28]MHz.

Aliased ADC



• Aliased ADC





• FM Demodulator




• Summary






Need an analogue bandpass filter to extract the FM band.

Aliased ADC



• Aliased ADC





• FM Demodulator




• Summary






Need an analogue bandpass filter to extract the FM band. Transition bandmid-points are at fs = 80MHz and 1.5fs = 120MHz.

Aliased ADC



• Aliased ADC





• FM Demodulator




• Summary







You can use an aliased analog-digital converter (ADC) provided that thetarget band fits entirely between two consecutive multiples of 1

2fs.

Aliased ADC



• Aliased ADC





• FM Demodulator




• Summary







You can use an aliased analog-digital converter (ADC) provided that thetarget band fits entirely between two consecutive multiples of 1

2fs.Lower ADC sample rate ,. Image = undistorted frequency-shifted copy.

Channel Selection



• Aliased ADC





• FM Demodulator




• Summary


FM band shifted to 7.5 to 28MHz (from 87.5 to 108MHz)

Channel Selection



• Aliased ADC





• FM Demodulator




• Summary



We need to select a single channel 200 kHz wide

Channel Selection



• Aliased ADC





• FM Demodulator




• Summary




We shift selected channel to DC and then downsample to fs = 400 kHz.Assume channel centre frequency is fc = c× 100 kHz

Channel Selection



• Aliased ADC





• FM Demodulator




• Summary





We must apply a filter before downsampling to remove unwanted images

Channel Selection



• Aliased ADC





• FM Demodulator




• Summary






The downsampled signal is complex since positive and negativefrequencies contain different information.

Channel Selection



• Aliased ADC





• FM Demodulator




• Summary







We will look at three methods:1 Freq shift, then polyphase lowpass filter

Channel Selection



• Aliased ADC





• FM Demodulator




• Summary







We will look at three methods:1 Freq shift, then polyphase lowpass filter2 Polyphase bandpass complex filter

Channel Selection



• Aliased ADC





• FM Demodulator




• Summary







We will look at three methods:1 Freq shift, then polyphase lowpass filter2 Polyphase bandpass complex filter3 Polyphase bandpass real filter

Channel Selection (1)



• Aliased ADC





• FM Demodulator




• Summary


Multiply by e−j2πr fc80MHz to shift

channel at fc to DC.fc = c× 100 k ⇒ fc

80M = c800




• Aliased ADC





• FM Demodulator




• Summary




80M = c800

Result of multiplication is complex(thick lines on diagram)




• Aliased ADC





• FM Demodulator




• Summary




80M = c800


Next, lowpass filter to ±100 kHz∆ω = 2π 200 k

80 M = 0.157




• Aliased ADC





• FM Demodulator




• Summary




80M = c800



80 M = 0.157

⇒ M = 60 dB3.5∆ω

= 1091




• Aliased ADC





• FM Demodulator




• Summary




80M = c800



80 M = 0.157

⇒ M = 60 dB3.5∆ω

= 1091

Finally, downsample 200 : 1




• Aliased ADC





• FM Demodulator




• Summary




80M = c800



80 M = 0.157

⇒ M = 60 dB3.5∆ω

= 1091


Polyphase:Hp(z) has

⌈

1092200

⌉

= 6 taps




• Aliased ADC





• FM Demodulator




• Summary




80M = c800



80 M = 0.157

⇒ M = 60 dB3.5∆ω

= 1091


Polyphase:Hp(z) has

⌈

1092200

⌉

= 6 taps

Complex data × Real Coefficients (needs 2 multiplies per tap)




• Aliased ADC





• FM Demodulator




• Summary




80M = c800



80 M = 0.157

⇒ M = 60 dB3.5∆ω

= 1091


Polyphase:Hp(z) has

⌈

1092200

⌉

= 6 taps

Complex data × Real Coefficients (needs 2 multiplies per tap)

Multiplication Load:2× 80MHz (freq shift) + 12× 80MHz (Hp(z)) = 14× 80MHz




• Aliased ADC





• FM Demodulator




• Summary


Channel centre frequency fc = c× 100 kHz where c is an integer.




• Aliased ADC





• FM Demodulator




• Summary


Channel centre frequency fc = c× 100 kHz where c is an integer.Write c = 4k + l

where k =⌊

c4

⌋

and l = cmod 4




• Aliased ADC





• FM Demodulator




• Summary



where k =⌊

c4

⌋

and l = cmod 4




• Aliased ADC





• FM Demodulator




• Summary



where k =⌊

c4

⌋

and l = cmod 4

We multiply u[r] by e−j2πr c800 , convolve with h[m] and then downsample:

v[n] =∑M

m=0 h[m]u[200n−m]e−j2π(200n−m) c800 [r = 200n]




• Aliased ADC





• FM Demodulator




• Summary



where k =⌊

c4

⌋

and l = cmod 4


v[n] =∑M

m=0 h[m]u[200n−m]e−j2π(200n−m) c800 [r = 200n]

=∑M

m=0 h[m]ej2πmc800u[200n−m]e−j2π200n 4k+l

800 [c = 4k + 1]




• Aliased ADC





• FM Demodulator




• Summary



where k =⌊

c4

⌋

and l = cmod 4


v[n] =∑M

m=0 h[m]u[200n−m]e−j2π(200n−m) c800 [r = 200n]

=∑M


800 [c = 4k + 1]

=∑M

m=0 g[c][m]u[200n−m]e−j2π ln4 [g[c][m]

∆= h[m]ej2π

mc800 ]




• Aliased ADC





• FM Demodulator




• Summary



where k =⌊

c4

⌋

and l = cmod 4


v[n] =∑M

m=0 h[m]u[200n−m]e−j2π(200n−m) c800 [r = 200n]

=∑M


800 [c = 4k + 1]

=∑M


∆= h[m]ej2π

mc800 ]

= (−j)ln∑M

m=0 g[c][m]u[200n−m] [e−j2π ln4 indep of m]




• Aliased ADC





• FM Demodulator




• Summary



where k =⌊

c4

⌋

and l = cmod 4


v[n] =∑M

m=0 h[m]u[200n−m]e−j2π(200n−m) c800 [r = 200n]

=∑M


800 [c = 4k + 1]

=∑M


∆= h[m]ej2π

mc800 ]

= (−j)ln∑M





• Aliased ADC





• FM Demodulator




• Summary



where k =⌊

c4

⌋

and l = cmod 4


v[n] =∑M

m=0 h[m]u[200n−m]e−j2π(200n−m) c800 [r = 200n]

=∑M


800 [c = 4k + 1]

=∑M


∆= h[m]ej2π

mc800 ]

= (−j)ln∑M


Multiplication Load for polyphase implementation:G[c],p(z) has complex coefficients × real input ⇒ 2 mults per tap




• Aliased ADC





• FM Demodulator




• Summary



where k =⌊

c4

⌋

and l = cmod 4


v[n] =∑M

m=0 h[m]u[200n−m]e−j2π(200n−m) c800 [r = 200n]

=∑M


800 [c = 4k + 1]

=∑M


∆= h[m]ej2π

mc800 ]

= (−j)ln∑M



(−j)ln

∈ {+1, −j, −1, +j} so no actual multiplies needed




• Aliased ADC





• FM Demodulator




• Summary



where k =⌊

c4

⌋

and l = cmod 4


v[n] =∑M

m=0 h[m]u[200n−m]e−j2π(200n−m) c800 [r = 200n]

=∑M


800 [c = 4k + 1]

=∑M


∆= h[m]ej2π

mc800 ]

= (−j)ln∑M



(−j)ln

∈ {+1, −j, −1, +j} so no actual multiplies neededTotal: 12× 80MHz (for G[c],p(z)) + 0 (for −jln) = 12× 80MHz




• Aliased ADC





• FM Demodulator




• Summary


Channel frequency fc = c× 100 kHz where c = 4k + l is an integer




• Aliased ADC





• FM Demodulator




• Summary



g[c][m] = h[m]ej2πcm800




• Aliased ADC





• FM Demodulator




• Summary




g[c],p[s] = gc[200s+ p]= h[200s+ p]ej2πc(200s+p)

800 [polyphase]




• Aliased ADC





• FM Demodulator




• Summary





800 [polyphase]

= h[200s+ p]ej2πcs4 ej2π

cp800




• Aliased ADC





• FM Demodulator




• Summary





800 [polyphase]


cp800 , h[200s+ p]ej2π

cs4 αp




• Aliased ADC





• FM Demodulator




• Summary





800 [polyphase]


cp800 , h[200s+ p]ej2π

cs4 αp

Define f[c],p[s] = h[200s+ p]ej2π(4k+l)s

4 = jlsh[200s+ p]




• Aliased ADC





• FM Demodulator




• Summary





800 [polyphase]


cp800 , h[200s+ p]ej2π

cs4 αp


4 = jlsh[200s+ p]

Although f[c],p[s] is complex it requires only one multiplication pertap because each tap is either purely real or purely imaginary.




• Aliased ADC





• FM Demodulator




• Summary





800 [polyphase]


cp800 , h[200s+ p]ej2π

cs4 αp


4 = jlsh[200s+ p]

Although f[c],p[s] is complex it requires only one multiplication pertap because each tap is either purely real or purely imaginary.

Multiplication Load:6× 80 MHz (Fp(z)) + 4× 80MHz (×ej2π

cp800 ) = 10× 80MHz

FM Demodulator



• Aliased ADC





• FM Demodulator




• Summary


Complex FM signal centred at DC: v(t) = |v(t)|ejφ(t)

FM Demodulator



• Aliased ADC





• FM Demodulator




• Summary



We know that log v = log |v|+ jφ

FM Demodulator



• Aliased ADC





• FM Demodulator




• Summary




The instantaneous frequency of v(t) is dφdt

.

FM Demodulator



• Aliased ADC





• FM Demodulator




• Summary





.

We need to calculate x(t) = dφdt

= dℑ(log v)dt

FM Demodulator



• Aliased ADC





• FM Demodulator




• Summary





.


= dℑ(log v)dt

= ℑ(

1vdvdt

)

FM Demodulator



• Aliased ADC





• FM Demodulator




• Summary





.


= dℑ(log v)dt

= ℑ(

1vdvdt

)

= 1|v|2ℑ

(

v∗ dvdt

)

FM Demodulator



• Aliased ADC





• FM Demodulator




• Summary





.


= dℑ(log v)dt

= ℑ(

1vdvdt

)

= 1|v|2ℑ

(

v∗ dvdt

)

FM Demodulator



• Aliased ADC





• FM Demodulator




• Summary





.


= dℑ(log v)dt

= ℑ(

1vdvdt

)

= 1|v|2ℑ

(

v∗ dvdt

)

We need:(1) Differentiation filter, D(z)

FM Demodulator



• Aliased ADC





• FM Demodulator




• Summary





.


= dℑ(log v)dt

= ℑ(

1vdvdt

)

= 1|v|2ℑ

(

v∗ dvdt

)

We need:(1) Differentiation filter, D(z)(2) Complex multiply, w[n]× v∗[n] (only need ℑ part)

FM Demodulator



• Aliased ADC





• FM Demodulator




• Summary





.


= dℑ(log v)dt

= ℑ(

1vdvdt

)

= 1|v|2ℑ

(

v∗ dvdt

)

We need:(1) Differentiation filter, D(z)(2) Complex multiply, w[n]× v∗[n] (only need ℑ part)(3) Real Divide by |v|2

FM Demodulator



• Aliased ADC





• FM Demodulator




• Summary





.


= dℑ(log v)dt

= ℑ(

1vdvdt

)

= 1|v|2ℑ

(

v∗ dvdt

)

We need:(1) Differentiation filter, D(z)(2) Complex multiply, w[n]× v∗[n] (only need ℑ part)(3) Real Divide by |v|2

x[n] is baseband signal (real):

Differentiation Filter



• Aliased ADC





• FM Demodulator




• Summary





• Aliased ADC





• FM Demodulator




• Summary


Window design method:(1) calculate d[n] for the ideal filter(2) multiply by a window to give finite support




• Aliased ADC





• FM Demodulator




• Summary



Differentiation: ddtejωt = jωejωt




• Aliased ADC





• FM Demodulator




• Summary



Differentiation: ddtejωt = jωejωt ⇒ D(ejω) =

{

jω |ω| ≤ ω0

0 |ω| > ω0




• Aliased ADC





• FM Demodulator




• Summary




{

jω |ω| ≤ ω0

0 |ω| > ω0

Hence d[n] = 12π

∫ ω0

−ω0jωejωndω = j

2π

[

ωejnω

jn− ejnω

j2n2

]ω0

−ω0

[IDTFT]

= nω0 cosnω0−sinnω0

πn2




• Aliased ADC





• FM Demodulator




• Summary




{

jω |ω| ≤ ω0

0 |ω| > ω0

Hence d[n] = 12π

∫ ω0


2π

[

ωejnω

jn− ejnω

j2n2

]ω0

−ω0

[IDTFT]


πn2

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

ω (rad/sample)

|H|

ω0

0 0.5 1 1.5 2 2.5 3

-80

-60

-40

-20

0

ω (rad/sample)

|H| (

dB)

ω0

Using M = 18, Kaiser window, β = 7 and ω0 = 2.2 = 2π×140 kHz400 kHz :




• Aliased ADC





• FM Demodulator




• Summary




{

jω |ω| ≤ ω0

0 |ω| > ω0

Hence d[n] = 12π

∫ ω0


2π

[

ωejnω

jn− ejnω

j2n2

]ω0

−ω0

[IDTFT]


πn2

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

ω (rad/sample)

|H|

ω0

0 0.5 1 1.5 2 2.5 3

-80

-60

-40

-20

0

ω (rad/sample)

|H| (

dB)

ω0


Near perfect differentiation for ω ≤ 1.6 (≈ 100 kHz for fs = 400 kHz)




• Aliased ADC





• FM Demodulator




• Summary




{

jω |ω| ≤ ω0

0 |ω| > ω0

Hence d[n] = 12π

∫ ω0


2π

[

ωejnω

jn− ejnω

j2n2

]ω0

−ω0

[IDTFT]


πn2

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

ω (rad/sample)

|H|

ω0

0 0.5 1 1.5 2 2.5 3

-80

-60

-40

-20

0

ω (rad/sample)

|H| (

dB)

ω0


Near perfect differentiation for ω ≤ 1.6 (≈ 100 kHz for fs = 400 kHz)Broad transition region allows shorter filter

Pilot tone extraction +



• Aliased ADC





• FM Demodulator




• Summary


Aim: extract 19 kHz pilot tone, double freq → real 38 kHz tone.




• Aliased ADC





• FM Demodulator




• Summary






• Aliased ADC





• FM Demodulator




• Summary



(1) shift spectrum down by 20 kHz: multiply by e−j2πn 20kHz400kHz




• Aliased ADC





• FM Demodulator




• Summary




(2) low pass filter to ±1 kHz to extract complex pilot at −1 kHz: H(z)




• Aliased ADC





• FM Demodulator




• Summary





(3) square to double frequency to −2 kHz [(

ejωt)2

= ej2ωt]




• Aliased ADC





• FM Demodulator




• Summary






ejωt)2

= ej2ωt]

(4) shift spectrum up by 40 kHz: multiply by e+j2πn 40kHz400 kHz




• Aliased ADC





• FM Demodulator




• Summary






ejωt)2

= ej2ωt]


(5) take real part




• Aliased ADC





• FM Demodulator




• Summary






ejωt)2

= ej2ωt]


(5) take real part

More efficient to do low pass filtering at a low sample rate:




• Aliased ADC





• FM Demodulator




• Summary






ejωt)2

= ej2ωt]


(5) take real part


Transition bands:F (z): 1 → 17 kHz, H(z): 1 → 3 kHz




• Aliased ADC





• FM Demodulator




• Summary






ejωt)2

= ej2ωt]


(5) take real part


Transition bands:F (z): 1 → 17 kHz, H(z): 1 → 3 kHz, G(z): 2 → 18 kHz




• Aliased ADC





• FM Demodulator




• Summary






ejωt)2

= ej2ωt]


(5) take real part


Transition bands:F (z): 1 → 17 kHz, H(z): 1 → 3 kHz, G(z): 2 → 18 kHz

∆ω = 0.25 ⇒ M = 68, ∆ω = 0.63 ⇒ 27, ∆ω = 0.25 ⇒ 68

Polyphase Pilot tone



• Aliased ADC





• FM Demodulator




• Summary





• Aliased ADC





• FM Demodulator




• Summary





• Aliased ADC





• FM Demodulator




• Summary


Anti-alias filter: F (z)Each branch, Fp(z), gets every 20th sample and an identical ej2π

n20




• Aliased ADC





• FM Demodulator




• Summary



n20

So Fp(z) can filter a real signal and then multiply by fixed ej2πp20




• Aliased ADC





• FM Demodulator




• Summary



n20


Anti-image filter: G(z)Each branch, Gp(z), multiplied by identical ej2π

n10




• Aliased ADC





• FM Demodulator




• Summary



n20



n10

So Gp(z) can filter a real signal




• Aliased ADC





• FM Demodulator




• Summary



n20



n10


Multiplies:F and G each: (4 + 2)× 400 kHz




• Aliased ADC





• FM Demodulator




• Summary



n20



n10


Multiplies:F and G each: (4 + 2)× 400 kHz, H + x2: (2× 28 + 4)× 20 kHz




• Aliased ADC





• FM Demodulator




• Summary



n20



n10


Multiplies:F and G each: (4 + 2)× 400 kHz, H + x2: (2× 28 + 4)× 20 kHzTotal: 15× 400 kHz




• Aliased ADC





• FM Demodulator




• Summary



n20



n10


Multiplies:F and G each: (4 + 2)× 400 kHz, H + x2: (2× 28 + 4)× 20 kHzTotal: 15× 400 kHz [Full-rate H(z) needs 273× 400 kHz]

Summary



• Aliased ADC





• FM Demodulator




• Summary


• Aliased ADC allows sampling below the Nyquist frequency◦ Only works because the wanted signal fits entirely within a

Nyquist band image

Summary



• Aliased ADC





• FM Demodulator




• Summary



Nyquist band image

• Polyphase filter can be combined with complex multiplications toselect the desired image◦ subsequent multiplication by −jln shifts by the desired multiple

of 14 sample rate

⊲ No actual multiplications required

Summary



• Aliased ADC





• FM Demodulator




• Summary



Nyquist band image


of 14 sample rate


• FM demodulation uses a differentiation filter to calculate dφ

dt

Summary



• Aliased ADC





• FM Demodulator




• Summary



Nyquist band image


of 14 sample rate



dt

• Pilot tone bandpass filter has narrow bandwidth so better done at alow sample rate◦ double the frequency of a complex tone by squaring it

Summary



• Aliased ADC





• FM Demodulator




• Summary



Nyquist band image


of 14 sample rate



dt

• Pilot tone bandpass filter has narrow bandwidth so better done at alow sample rate◦ double the frequency of a complex tone by squaring it

This example is taken from Harris: 13.

Date post:	13-Jan-2017
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14: FM Radio Receiver

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