However, had it been a real sound, the sound's pitch would have been increased by the doppler effect, since the falcon was moving the source of the sound. perpendicular to away from towards at the same speed as

Towards. The Doppler Effect tells us that as an object moves toward the source of sound, the pitch the object feels will increase; when the object moves away, the pitch decreases. In order for the pitch to increase, the falcon must be moving towards the source of sound.